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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Is $f(x,y)=g(x \cdot y) $ continuous?

Let $f(x, y) = g(xy)$ where $g :\mathbb{R}\to \mathbb{R}$ is continuous. Is $f$ continuous? Proof: Consider $H(x)=x$ and $Z(y)=y$ both of these functions are continuous so is their product $K(x,y)=x ...
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79 views

Does a function have to be bounded to be uniformly continuous?

My book defines uniform continuity as a form of continuity that works for any points $a$ and $x$ in an interval $I$ such that $$|x-a| < \delta$$ implies that $$f(x) - f(a) < \epsilon$$ It then ...
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46 views

Supremum of a function

I stumbled upon this question from a previous exam in my Real analysis course. Let $f \colon (0, 1] \to [0, 2]$ defined by $f (x) = 1 + (1 − x) \sin (1 /x)$ Prove that $\sup f((0,1]) = 2$. Prove ...
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3answers
107 views

Is $f:\Bbb Q\rightarrow \Bbb R$ continuous?

Define $f:Q\rightarrow R$ such that $f= 1$ if $x^2<2$ or $f=0$ is $x^2>2 $. Is this function continuous? All proofs are welcomed but i am more interested using the definitions maybe the ...
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$\int_{a}^{b}f'=f(b)-f(a)=>\int_{a}^{b}\lim_{n->\infty}Diff_{\frac{1}{n}}f=\lim_{n->\infty}\int_{a}^{b}Diff_{\frac{1}{n}}f$

I want to show that if $f$ is continuous on $[a,b]$ and differentiable a.e. on $(a,b)$. Then $\int_{a}^{b}f'=f(b)-f(a)=>\int_{a}^{b}\lim_{n->\infty}Diff_{\frac{1}{n}}f=\lim_{n->\infty}\int_{...
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Understanding the notation $f\colon D\subseteq\mathbb{R}\to\mathbb{R}$ in the context of uniform continuity

In brief, I've had virtually no mathematical education prior to seven months ago. What I did know had largely been the result of watching Khan Academy. Seven months ago I began a precalculus textbook, ...
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42 views

A a criterion for continuity

Let $(X, \mathcal{A})$ be a topological space. Prove that a real-valued function $f$ on $X$ is continuous if and only if for every real number $a$ the sets $\{x \in X: f(x) < a \}$ and $\{x \in X: ...
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1answer
19 views

$f$ has zero of multiplicity $m$ at $\alpha$ and $k$ at $\beta$, where $m+k-1=n$. Prove that $f^{(n)}$ has at least one zero in $(\alpha, \beta)$.

Let $f\in C^n[a,b]$. Suppose that $f$ has a zero of multiplicity $m$ at $\alpha$ and a root of multiplicity $k$ at $\beta$, where $m\geq1$, $k\geq1$, and $m+k-1=n$. Prove that $f^{(n)}$ has at least ...
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1answer
44 views

Does it matter how small $\epsilon$ is?

If I'm trying to prove that a function is continuous at a certain point, and through my argument I needed that $\epsilon < \alpha$, where $\alpha$ is some known positive constant (say $1$ or $10^{-...
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67 views

Given $A$ a symmetric square matrix. Why is $f_A: O(n) \to \mathbb{R}^{n \times n}: Q \mapsto Q^{\top} A Q$ continuous?

Given $A$ a symmetric square matrix. And $O(n)$ the set of orthogonal matrices. Why is the following function continuous: $$f_A: O(n) \to \mathbb{R}^{n \times n}: Q \mapsto Q^{\top} A Q$$ This is ...
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19 views

Weak contraction mapping preserves order?

Suppose I have a function $\varphi$ on the unit interval such that $|\varphi (x)-\varphi (y)|<|x-y|$. Is it true that if $0 \leqslant c \leqslant 1$ then $\varphi (0) \leqslant \varphi (c) \...
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1answer
78 views

Reconstructing a function from its critical points and inflection points

For my class I have sketched the following graph which depicts certain critical points and curvatures including one local maximum, minimum, two inflection points and a saddle point (which is an ...
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2answers
24 views

Prove that $f$ is continuous over $\cup _{i=1}^m F_i$ [duplicate]

Problem: Let $f:X\to Y$ and let $F_i$ be a collection of closed sets such that $f|_{F_i}$ is continuous for each $i$ . Prove that $f$ is continuous over $\cup_{i=1}^m F_i$ ;$X,Y$ are metric spaces....
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38 views

Why does a Poisson process hurt the prerequesites of the Kolmogorov-Chentsov theorem

I have a question for you. Obviously, by looking at the sample paths of a Poisson process with parameter $\lambda >0$, this process does not have a hölder-continuous version. But why? I have the ...
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1answer
81 views

Prove symmetry of probabilities given random variables are iid and have continuous cdf

Let $Y_1, Y_2, ...$ be independent and identically distributed random variables in $(\Omega, \mathscr{F}, \mathbb{P})$ s.t. their distributions are continuous and $$F_{Y}(y) := F_{Y_1}(y) = F_{Y_2}(y) ...
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1answer
31 views

$f \in C^2(0,\infty)$ , then how to show that $\Big(\sup_{x>0} |f'(x)|\Big)^2\le 4\big(\sup _{x>0}|f(x)|\big)\big(\sup_{x>0}|f''(x)|\big)$ ? [duplicate]

Let $f \in C^2(0,\infty)$ , then how to show that $\Big(\sup_{x>0} |f'(x)|\Big)^2\le 4\big(\sup _{x>0}|f(x)|\big)\big(\sup_{x>0}|f''(x)|\big)$ ?
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1answer
26 views

Creating a monotonous rising function that is not equivalent to a continuous function on any interval

I have the answer given. I do not understand only one step in it. $\mathbb Q$ is countable, so therefore we can put it's points in a array: $\{q_0,q_1,...\}$ and then the function $$f(x)=\sum_{\...
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1answer
36 views

The continuous relationship between $f(x,y)$ and $\varphi(x)=\lim f(x,y)$

Assume $\Bbb{D}=\{(x,y):a\leq x \leq b,0 \leq y <y_0\}$ and $f(x,y)$ is continuous for variable $x$ in $\Bbb{D}$. and $f(x,y)$ increases monotonically to $\varphi(x)$ when $y \rightarrow y_0$. ...
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68 views

A two-to-one real function must have infinitely many points of discontinuity [closed]

If a function $f:\mathbb{R}\to\mathbb{R}$ attains each value twice then prove that it has infinitely many points of discontinuity.
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41 views

If $f:[0,\infty) \to \mathbb{R}$ is continuous and $f(x) > 0$ for all $x$, then there exists $c > 0$ such that $f(x)\geq c$ for all $x$.

If $f:[0,\infty) \to \mathbb{R}$ is continuous and $f(x) > 0$ for all $x$, then there exists $c > 0$ such that $f(x)\geq c$ for all $x$. I feel that this is a false statement given the order of ...
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1answer
58 views

If $f:[0,1) \to \mathbb{R}$ is continuous and $f(x) > 0$ for all $x$, then there exists $c > 0$ such that $f(x)\geq c$ for all $x$.

If $f:[0,1) \to \mathbb{R}$ is continuous and $f(x) > 0$ for all $x$, then there exists $c > 0$ such that $f(x)\geq c$ for all $x$. I feel that this is a false statement given the order of the ...
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1answer
149 views

Composition of lower semicontinuous function with continuous function is lower semicontinuous

Assume that $f\colon \mathbb{R}^n \to\mathbb{R}$ is lower semicontinuous at $g(a)$ and $g\colon \mathbb{R}^m \to\mathbb{R}^n$ is continuous at $a \in\mathbb{R}$. Define $h = f \circ g \colon \mathbb{...
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32 views

Confusion about Differentiability with partial derivatives

Theorem: Let $A ⊂ R^n$ be open and $f: A ⊂ R^n → R^m$. Suppose $f = (f_1,...,f_m)$. If each of the partials $\frac{∂f_j}{∂x_i}$ exists and is continuous on $A$, then $f$ is differentiable on A. ...
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45 views

A Function is indefinitely differentiable

Suppose $f(x)=e^{-\frac{1}{x-a}-\frac{1}{x-b}}$ if $x \in (a,b)$, and $f(x)=0$ otherwise, show that $f \in C_{c}^{\infty} (R)$ I think I have to just find the general form of the $n^{th}$ derivative ...
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33 views

Prove or find a counter example to those statements.

Let the funktions $f,g:\mathbb{R} \rightarrow \mathbb{R}$ and $a\in\mathbb{R}$. a) $f$ is continuos in $a$ $\Leftrightarrow$ $|f|$ is continuos in a; b) $f,g$ are continuos in $a$ $\Rightarrow$ $max\...
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1answer
38 views

If an operator is weak$^*$-to-weak$^*$ continuous, will it induce an operator?

Suppose that $X$ and $Y$ are Banach spaces and have predual, say $X_*$ and $Y_*$. Define an operator $T:X \rightarrow Y$. If $T$ is weak$^*$-to-weak$^*$ continuous, will $T$ induce an operator $T_* :...
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Does local Lipschitz continuity on a compact metric space imply global Lipschitz continuity? [duplicate]

I've got two problems on this issue: If $X,Y$ are metric spaces, $X$ compact, suppose $f:X\to Y$ is locally Lipschitz continuous on $X$, then is it also globally Lipschitz continuous? My guess ...
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1answer
87 views

Show that the derivative of a function is not continuous

$$g(x)=\begin{cases} x+2x^2\sin\left(\frac{1}{x}\right)&\text{ if }x\neq0\\\ 0&\text{ if }x=0 \end{cases}$$ Show that there is a sequence $\{x_n\}$ with $\{x_n\} \to 0$ as $n$ approaches ...
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64 views

The composition of a lower semicontinuous function with a continuous one is lower semicontinuous

Assume that $f: \mathbb{R}^{n} \to \mathbb{R}$ is lower semicontinuous at $g(a)$ and $g : \mathbb{R}^{m} \to \mathbb{R}^{n}$ is continuous at $a \in \mathbb{R}^{n}$. Define $h = f \circ g : \mathbb{R}^...
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1answer
39 views

Continuity and Differentiable functions

Let $f$ be continuous and differentiable on the interval $[a, b]$. Assuming $f$ is bounded on the interval $[a, b]$ and $m = \inf\limits_{[a,b]} f(x)$, prove that there exists $d \in [a, b]$ such that ...
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32 views

Using $\epsilon$-$\delta$ to show a that continuous function whose image contains only rationals is constant.

Suppose $f:[0,1]\to\mathbb{R}$ is continuous and it's image contains only rationals, then $f$ is constant. This can be easily shown using the Intermediate Value Theorem. But how about trying to use ...
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87 views

Limit of reciprocal function is the reciprocal of its limit

Let $g$ be continuous on the interval $[a, b]$, and a $< c < b$ with $g_{(c)}<0$ Prove, using the epsilon and delta characterisation of a limit, that $\lim\limits_{x \to {c}}\frac{1}{g{(x)}} ...
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105 views

Proving that a one-to-one continuous function on a compact subset has a continuous inverse

This is a curious problem I found in the "challenge" section of the text I'm learning real analysis from. Suppose $A$ is a compact subset of $\mathbb{R}^{n}$ and that $f$ is a continuous function ...
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1answer
24 views

Is this a continuity / connectedness argument or is it an orientation-preservation argument?

Take, for example, the simple linear fractional transformation that sends the upper half plane to the unit disk, and the real line to the unit circle. We know the fact that the upper half plane (UHP) ...
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If a continuous function takes equal values at endpoints, in cannot be injective in the interior

Suppose $f$ is a nonconstant continuous function on $[0,2]$ and satisfies $f(0)=f(2)=0$. Show $f$ cannot be one to one on the open interval $(0,2)$ I'm a little confused about this question. If $f(0)=...
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Prove Piecewise Function Integrable

$$ f(x) = \begin{cases} -2, & \text{if }x < 0 \\ 1, & \text{if }x > 0\\ 0, & \text{if }x = 0 \end{cases} $$ Hey guys I need some help showing that this function is integrable on the ...
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proving continuity of a function from f(x+h) and f(x-h)

Let f be such that lim as h approaches 0 of: (f(x+h) - f(x-h)) = 0 for all real numbers x. Does this imply that f is continuous?
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Show that a subset $X$ of $M$ is open iff $X$ = $f^{-1}$($V$) for some continuous function $f$: $M$ into $R$ and some open subset $V$ of $R$

Show that a subset $X$ of $M$ is open iff $X$ = $f^{-1}$($V$) for some continuous function $f$: $M$ into $R$ and some open subset $V$ of $R$ I'm having a bit of trouble of doing this problem.
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50 views

Show that $F(x) = \int_{-\infty ,x}f$ is continuous on $\mathbb{R}$

Let $f \in L^1(m)$ where $m$ is the standard Lebesgue measure. Show that $$F(x) = \int_{-\infty}^{x}f \text{ is continuous on } \mathbb{R}.$$ I think what is confusing me is that my professor labeled ...
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57 views

Prove continuity by the epsilon-delta definition

Let $a>0$, $b>0$ and $f:\mathbb{R} \to \mathbb{R}$ define by \begin{align} f(x) = \left\{ \begin{array}{ll} x^a \sin{x^{-b}} \quad &\text{if } x \ne 0, \\ 0 \quad &\text{if } x = 0 \end{...
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19 views

Prove that Urysohn spaces are completely Hausdorff

Theorem: Urysohn spaces are $T_{2\frac{1}{2}}$. My attempt: Let $(X, \tau)$ be an Urysohn space. Let $u, v$ be distinct points in $X$. Let $f$ be an Urysohn function for $\{u\}$ and $\{v\}$. ...
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Rudin's Principles of Mathematical Analysis Ch 4 Exercise 14 [duplicate]

So, as part of a homework assignment my professor assigned me the following exercise from Rudin's Principles of Mathematical Analysis: Let $I = [0,1]$ be the closed unit interval. Suppose $f$ is ...
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29 views

How to find the preimage of a set

I'm trying to prove continuity of two different functions $f\colon A\to B$. I know that to find continuity, the inverse of all open subsets in $B$ must be open in $A$. I also know that to find the ...
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54 views

Discontinuous function that admits antiderivatives.

Given the function $g$ defined on $[-1, 1]$ with real values, having a plot as depicted in the image, can you prove that $g$ has antiderivatives on $[-1, 1]$? All the triangles are isosceles and are ...
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19 views

What is unilateral continuity?

I wanted to ask if anyone knows what unilateral continuity is. In particular, I am considering the multiplication map of group from $G\times G$ to $G$. This is being used in one of the papers I am ...
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2answers
47 views

Show that $f$ is discontinuous at $c \in (a, b)$ if and only if …

I would like to prove the following: "Suppose $f$ is a bounded function on $[a, b]$. Show that $f$ is discontinuous at $c \in (a,b)$ if and only if $\exists n \in \mathbb{N}$ s.t. $$\sup_{x \in I}f(x)...
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29 views

Closed set mapped to a not closed example

Give an example for the following cases: A closed set $S \in \Bbb R$ and a continuous function $f: \Bbb R \to \Bbb R,$ such that $f(S)$ is not closed. A bounded set $S \in \Bbb R \setminus0$ and a ...
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47 views

A continuous selection from a countable family of uniformly Lipschitz functions is Lipschitz

Conider a countable collection of positive Lipschitz functions $\varphi_n : \mathbb{R} \to \mathbb{R}$ such that there is a positive real number $K$ which works as a Lipschitz constant for every $\...
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31 views

Maximum of two $C^2$ functions

Consider two $C^2$ functions $f, g : \mathbb{R} \to \mathbb{R}$ such that their derivatives satisfy $|\frac{df}{dx}| < e^{-x}$ and $|\frac{dg}{dx}| < e^{-x}$. Now I want to look at the ...
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1answer
49 views

Proof of continuity at $x=0$

Prove $f$ is continuous at $0$, when $f(x)$ is defined as $$f(x)=\begin{cases}x^3\cos(1/x) &\mbox{if $x \ne 0$,} \\ 0 &\mbox{if $x=0$.} \end{cases}$$ Using delta-epsilon proof, $$|f(x)-f(s)| ...