Intuitively, a continuous function is one where small changes of input result in correspondingly small changes of output. Use this tag for questions involving this concept. As there are many mathematical formalizations of continuity, please also use an appropriate subject tag such as (real-analysis) ...

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50 views

Whether a continuous function has fixed point or not when the domain and range are not $[0,1]$

Which of the following is false $?$ $A.$ Any continuous function from $[0,1]$ to $[0,1]$ has a fixed point. $B.$ Any homeomorphism from $[0,1)$ to $[0,1)$ has a fixed point. $C.$ Any bounded ...
1
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1answer
32 views

Which Properties of a Natural Cubic Spline does the following function possess and not possess

I need to determine which of the properties of a natural cubic spline the following function possesses or does not possess: $$f(x) = \begin{cases} (x+1)+(x+1)^{3}, & x \in [-1,0] \\ ...
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1answer
55 views

$f:\mathbb R\rightarrow [0,\infty )$ is continuous such that $g(x)={(f(x))}^2$ is uniformly continuous .

$f:\mathbb R\rightarrow [0,\infty )$ is continuous such that $g(x)={(f(x))}^2$ is uniformly continuous . Then which of the following is always true $?$ $A.$ $f$ is bounded. $B.$ $f$ may not be ...
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34 views

The points where function is discontinuous,are those points counted/considered in the domain of the function.

The points where function is discontinuous,are those points counted/considered in the domain of the function. $(1)[x]$,greatest integer function is discontinuous at all integer points but integers are ...
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0answers
36 views

Is it possible to visit all points on a differentiable function by “rolling”?

I recall from a discussion thread some week ago we talked about different ways to pedagogically explain differentiability. So I came up with this idea that if there for each point there exists a ...
1
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1answer
34 views

Is the function $f(x)$ uniformly continuous on $(0, 1)$ if (a) $f(x) = x\sin(x^{-2})$ and if (b) $f(x) = \sin(x^{-2})$?

Is the function $f(x)$ uniformly continuous on $(0, 1)$ if (a) $f(x) = x\sin(x^{-2})$ (b) $f(x) = \sin(x^{-2})$ I have been using big O notation to try to solve this - since $$\sin(u) = u- ...
0
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4answers
38 views

Intermediate value theorem for two functions

Let $f: [a,b] \to \mathbb{R}$ and $g: [a,b] \to \mathbb{R}$ continuous with $a,b \in \mathbb{R}$. Then if $g(a) \leq f(a)$ and $f(b) \leq g(b)$, then it exists an $x \in [a,b]$ so that $f(x) = ...
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1answer
29 views

Show that the following operator (on a Hilbert space) is continuous.

"Let $\mathcal H$ be a complex Hilbert space and let $y\in\mathcal H.$ Show that the linear transformation $f:\mathcal H\to\mathbb C$ defined by, $f(x)=\langle x,y\rangle$ is continuous." ...
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1answer
40 views

What is the name of the theorem?

Let $f$ be continuous, $f(x_1)<0$ and $f(x_2)>0$, there has to be a root of $f$ between $x_1$ and $x_2$? I know it is very basic, but i was not using calculus for a long time, and name of ...
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0answers
81 views

Is it known whether the boundary of the Mandelbrot set is not continuous?

I might be missing something obvious here, but my understanding is that nobody currently knows whether the boundary of the Mandelbrot set is a Jordan curve because otherwise we would know that the ...
0
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2answers
20 views

Sigma-finiteness and absolutely continuous measures

Suppose we have an abstract $\sigma$-finite measure space $(X,\mathscr{A},\mu)$ and let $\nu$ be another measure on $(X,\mathscr{A})$ as well, with the property that $\nu \ll\mu$, i.e $\nu$ is ...
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1answer
112 views

Let $f: \Bbb R \to [0,\infty)$ be a continuous function such that $g(x)=(f(x))^2$ is uniformly continuous. Which of the following is always true?

A. $f$ is bounded B. $f$ may not be uniformly continuous C. $f$ is uniformly continuous D. $f$ is unbounded. Let $f(x)=\sqrt x$. Then $g(x)=x$ is uniformly continuous and ...
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4answers
81 views

$f:\mathbb R \to \mathbb R$ be differentiable such that $f(0)=0$ and $f'(x)>f(x),\forall x \in \mathbb R$ ; then is $f(x)>0,\forall x>0$?

Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $f(0)=0$ and $f'(x)>f(x),\forall x \in \mathbb R$ ; then is it true that $f(x)>0,\forall x>0$ ?
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2answers
68 views

Real analysis true/false

(1) Let $f$ be a continuous function on $[0,1]$. Then for every partition $P$ of $[0,1]$, the lower and upper Riemann sums of $f$ over $P$ satisfy $L(f,P)\neq U(f,P)$. (2) Let ...
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votes
3answers
92 views

Find the norm of the following operator.

Take $X=C([0,1])$ with the uniform norm, $\|f\|=\sup_{x\in[0,1]}|f(x)|$, and define the operator $T:X\to X$ by, $$T(f)(x)=f(x)-\int_0^1f(s)ds$$ Find $\|T\|$. I was hoping to solve this ...
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2answers
56 views

True/False: properties of a continuous map $f: S^1 \to \Bbb R$.

Let $S^1=\{z \in \Bbb C : |z|=1\}$. Let $f$ be any continuous function from $S^1 \to \Bbb R$. Then: A. $f$ is bounded B. $f$ is uniformly continuous C. $f$ has image containing a ...
5
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3answers
150 views

Multiple choice exercise on $f(x)= \frac {\sin x}{|x|+ \cos x}$

Let $f : \Bbb R \to \Bbb R$ be the function defined by $f(x)= \frac {\sin x}{|x|+ \cos x}$. Then A.$f$ is differentiable at all $x \in \Bbb R$. B.$f$ is not differentiable at $x =0$. ...
2
votes
2answers
77 views

Prove that $ f$ is uniformly continuous

Let $f:\Bbb R\to \Bbb R$ be a continuous function such that $f(i)=0\ \forall i\in \Bbb Z$ .Then is it true that $ f$ is uniformly continuous ? let $\epsilon >0$ be given.Then I have to find a ...
3
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3answers
138 views

Prove that $\lim f_n(x_n) = f(x), x_n\rightarrow x,$ then $f_n\rightarrow f$ uniformly on compact

From an exercise list: Prove that if a sequence of continuous functions $f_n:X\rightarrow \mathbb{R}$ is such that $x_n\in X$, $\lim x_n = x \in X \Rightarrow \lim f_n(x_n) = f(x)$, then ...
5
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3answers
123 views

Finding a unique continuous function

Let $f$ be a given continuous function on $[0,1]$. How do you prove that there is a unique continuous function $g$ on $[0,1]$ satisfying $$g(x) = \frac{1}{2}g\left(\frac{x+1}{2}\right) + f(x)$$ for ...
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1answer
39 views

$\frac{f(x)}{x}>c$ for strictly increasing function?

A complete description of the question is like this: $f$ is differentiable everywhere on $\mathbb{R}$ and $f'$ is continuous on $\mathbb{R}$. Also $f$ is strictly increasing. Given that $f(0) ...
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1answer
74 views

What does the continuity of $f'$ tell us about $f$?

Suppose $f$ is differentiable on $\mathbb{R}$ and its derivative $f'$ is continuous on the interval $[a,b]$. What constraints on $f$ would such condition give us?
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1answer
46 views

Is it possible to find such a function?

The subset S of $\mathbb{R}^2$ is defined by $S=\{(x,f(x))\}$, where $f:\mathbb{R}\rightarrow \mathbb{R} $ is defined everywhere but not continuous on $\mathbb{R}$. Is it possible to find such a ...
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2answers
43 views

Form functions that are continuous at one point in L^\infty a Banach space.

Is the subspace $\{f \in L^\infty(\mathbb{R}) ~|~ f \text{ is continuous at } x=0 \}$ a Banach space? The norm is of course the essential supremum. Does the essential supremum even notice a single ...
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2answers
40 views

Prove that if the complex function $|f(z)|^2$ is constant in $D$ and $f(z)$ is analytic in $D$, then $f(z)$ is constant in $D$.

My proof: Let $|f(z)|^2 = M$ for $z\in D$. Then $f(z) = \pm\sqrt{M}$ (not sure about this step, are there only two values for the square root of a complex number> No right? Could be more. But I ...
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1answer
46 views

Show $ f(x) < x\cdot f'(x)$ [closed]

Function $f:[0,\infty) \to \mathbb R$ continuous on $[0,\infty)$ and differentiable at $(0,\infty)$. $f(0)=0$. $f'$ is strictly increasing.
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1answer
116 views

Proof of “the continuous image of a connected set is connected”

None of the existing questions is exactly answering my question so I'm posting a new question, but feel free to refer me to some already answered question! In Rudin Theorem 4.22, we know that ...
2
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2answers
30 views

Define $S=\{(x,f(x))\}$ (a subset of $\mathbb{R}^2$ ), where $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous on $\mathbb{R}$.

Define $S=\{(x,f(x))\}$ (a subset of $\mathbb{R}^2$ ), where $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous on $\mathbb{R}$. I was trying to prove that $S$ is a closed set. It is clearly true by ...
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0answers
20 views

Define $f(x)=0$ if $x\in \mathbb{Q}^c$; $f(x)=p-q$ if $x\in \mathbb{Q}$ where $x=\frac{p}{q}$ in lowest terms

Define $f(x)=0$ if $x\in \mathbb{Q}^c$; $f(x)=p-q$ if $x\in \mathbb{Q}$ where $x=\frac{p}{q}$ in lowest terms (conventional way to represent rationals). The question asks that whether f is Riemann ...
0
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2answers
26 views

Continuous function and extreme value theorem.

Suppose $f(x)$ is a continued function defined on $[-1,1]$ with $f(0)>0$ and $f(\pm 1)=0$. Prove that there exist constants $a>0$ and $b$ such that $g(x)=-a\vert x\vert +b\geq f(x) (-1\leq x ...
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2answers
79 views

The function $f$ is Riemann integrable on $[-1,1]$. For any interval in $[-1,1]$, there are always both positive and negative values of $f(x)$.

The function $f$ is Riemann integrable on $[-1,1]$. For any interval in $[-1,1]$, there are always both positive and negative values of $f(x)$. How can I prove that $\int^1_{-1}f=0$? I think that ...
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1answer
42 views

Prove that $C = \{x \in X: (f_n(x)) \text{ converges} \}$ is a closed set in $X$

The question goes like: Let $X$ be a compact metric space and let ${f_n}$ be an equicontinuous sequence in $C(X)$. Show that $C = \{x \in X: (f_n(x)) \text{ converges} \}$ is a closed set in $X$ My ...
0
votes
1answer
12 views

Define $h(x)=f(x)$ if $x \in \mathbb{Q}$; $h(x)=-f(x)$ if $x \in \mathbb{Q}^c$.

Define $h(x)=f(x)$ if $x \in \mathbb{Q}$; $h(x)=-f(x)$ if $x \in \mathbb{Q}^c$, where $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous on $\mathbb{R}$. Given that $h(x)$ is continuous at $x=0$, ...
0
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1answer
28 views

How do I find explicit formula of function, examine continuity and draw their graphs?

Three-cylinder with height 4m and radii of the base 5, 3 and 1 m are going to put (in this order). Give an explicit formula for the following functions, you examine the functions on continuity and ...
4
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1answer
51 views

Obscure proof for $f\in \mathcal C_0(\mathbb R)\implies U_f\in u_f$ continuous.

Let $$\mathcal C_b(\mathbb R)=\{f:\mathbb R\longrightarrow \mathbb R: f\ \text{continuous and bounded}\}\quad \text{and}\quad \mathcal C_0(\mathbb R)=\{f\in\mathcal C_b(\mathbb R)\mid ...
2
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1answer
31 views

$f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous on the set $C \subseteq \mathbb{R}$

$f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous on the set $C \subseteq \mathbb{R}$ . The question asks that whether $cl(f(C)) \subseteq f(cl(C))$. ($cl$ indicates closure). I think this is true ...
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2answers
33 views

Suppose $g:\mathbb{R} \rightarrow \mathbb{R} $ and $h:\mathbb{R} \rightarrow \mathbb{R} $ are continuous on $\mathbb{R}$.

Suppose $g:\mathbb{R} \rightarrow \mathbb{R} $ and $h:\mathbb{R} \rightarrow \mathbb{R} $ are continuous on $\mathbb{R}$. $g(x)<h(x)$ for all $x\in [-1,1]$. Then there exists $\epsilon >0$ such ...
8
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1answer
108 views

Can it be that $f$ and $g$ are everywhere continuous but nowhere differentiable but that $f \circ g$ is differentiable?

So, I was just asking myself can something like this happen? I was thinking about some everywhere continuous but nowhere differentiable functions $f$ and $g$ and the natural question arose on can the ...
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1answer
20 views

Continuity of maps

We consider this topological space $(\mathbb{R},\tau)$ where: $$\tau=\{G\subset \mathbb{R}, (\mathbb{R}\setminus G)~\text{countable}~\}\cup\{\emptyset\}$$ and we consider the identity map $$f: ...
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1answer
19 views

Changing the order of limits

I'm looking for an example of a function $f(x,y)$ such that $$\lim_{x\to a}\{\lim_{y\to b} f(x,y)\}\neq \lim_{y\to b}\{\lim_{x\to a}f(x,y)\}$$
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1answer
33 views

Suppose that $f$ is continuous and that $g \circ f$ is differentiable. Must $f$ then be differentiable?

Suppose that $f$ is real function of a real variable defined on $(a,b)$ and that it is continuous on $(a,b)$. Suppose also that $g \circ f$ is differentiable on the set $f((a,b))$ and that $g$ is not ...
10
votes
5answers
118 views

$f$ continuous on $(a,b)$ and $|f|$ differentiable on $(a,b)$; is $f$ differentiable in $(a,b)$?

Let $f:(a,b) \to \mathbb R$ be a continuous function such that $|f|$ is differentiable in $(a,b)$ ; then is $f$ differentiable in $(a,b)$ ?
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2answers
48 views

Problem 12 at Section 5.3 in Bartle and Sherbert's books on Real Analysis

Consider the function $f(x) = \sup\{x^2, \cos x\}$, $x \in [0, \tfrac{\pi}{2}]$. Show that there exists a point $x_0 \in [0, \tfrac{\pi}{2}]$ such that $f(x_0) = \min_{x \in [0, \tfrac{\pi}{2}]} ...
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0answers
25 views

Differentiability of convex functions except at countably many points

There is this result in Notions of Convexity, Hormander. The relevant part of it reads: let $f$ be convex in an interval $I$ and $x$ be an interior point. Let $f_l'$ and $f_r'$ denote left derivative ...
4
votes
2answers
95 views

Using epsilon delta, prove the max function of continuous functions f,g is also continuous

First of all I know that this question has been asked already, but I'm looking for a proof simply using the definition of continuity ($\epsilon$, $\delta$) Suppose $f,g:D \to R$ are both continuous ...
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1answer
26 views

Prove that between every two roots of f there is a root of g and vice versa

Suppose $f$ and $g$ are two continuous and differentiable functions such that $f' = g$ and $g' = -f$. Prove that between every two consecutive roots of $f$, there is a root of $g$ and between every ...
1
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1answer
53 views

$f$ and $g$ are two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing .Then…

$f$ and $g$ are two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing .Then which of the following is true $?$ $A)$ If $g$ is continuous then so is $f\circ g$ counterexample : ...
1
vote
3answers
43 views

Is $p(x)=a\frac{x^2}{x}+b$ a linear polynomial?

My question at first, may seem too simple, Is $p(x)=a\frac{x^2}x+b$ a linear polynomial? But as we see the polynomial $p(x)$ is not continuous at $x=0$. So if it is not a polynomial, which ...
2
votes
3answers
95 views

Proving f(x)=0 for all x in [a,b] when we only know that f is continuous and f(x)=0 when x is rational. [duplicate]

The question is as follows a.) Let $f(x)$ be continuous function on an interval [a,b] and suppose that $f(x)=0$ for each rational value $x$ in [a,b]. Prove that $f(x) = 0$ for all $x \in [a,b]$. b.) ...
1
vote
1answer
31 views

Prove that if a property holds for a function on rationals and the function is continuous, the property must hold on all reals

The context in which the question arose was trying to prove that the exponential function is the unique continuous function with the property that $g(s + t) = g(s)g(t)$ for $s,t > 0$. I have shown ...