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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IVT question involving polynomial with even degree

Let $M(x)$ be an even polynomial with a positive leading coefficient, with $a_{2n} > 0, n\ge1 $. Show that there exists a constant $a*\in \mathbb{R}$ such that $M(x)+a = 0$ has a real root if ...
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18 views

How to define Square Root

I'm trying to understand how to define the square root of a complex function "globally". Let's say we have some function from some set $X$ onto $\mathbb{C} - \{0\}$: $$ f:X\to\mathbb{C}-\{0\} $$ and ...
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1answer
27 views

Prove for some $z_0 \in C$ the function $f(z)=|z-z_0|$ is continuous on all of $\mathbb{C}$

Let $z_0\in\mathbb{C}$ and $f(z)=|z-z_0|$. Show that $f$ is continuous on $\mathbb{C}$. I expect to see a proof using the triangle inequality. Note a function $f$ is continuous on $\mathbb{C}$ if ...
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1answer
31 views

How to prove this statement?

I cannot prove this proposition directly . Let $(X,d)$ and $(Y,d')$ be metrice spaces. Let $f$ be a function from $X$ to $Y$. If $\overline{f^{-1} ( B)} \subseteq f^{-1}( \overline B)$ for all ...
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1answer
14 views

Questions on continuously differentiable function on $[a,b]$

Let $f:[a,b]\rightarrow\mathbb{R}$ be a function. Normally we define derivatives of $f$ only at interior points in $[a,b]$. But when we write $f\in C^1([a,b])$, it means that $f$ is differentiable on ...
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Question about the application of continuous functions and IVT

I came across a question which says: Suppose that $f:[0,2 \pi] \to \mathbb{R}$ is continuous, and $f(0)=f(2 \pi)$. 1.Show that there exists $x \in [0,\pi]$ such that $f(x)=f(x+ \pi)$. ...
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2answers
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Finding all continuous functions so that $f^n(x)=x$ for some $n$.

I came up with this problem in class but I can't seem to solve it. I need to find all the functions $f$ with domain and codomain $\mathbb R$ such that there is an $n$ such that $f^n(x)=x$ for all $x$, ...
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2answers
35 views

Find the points where the function is continuous

Let $X \subset \mathbb{R}$ be a finite set and define $f: \mathbb{R} \to \mathbb{R}$ by $$ f(x)= \begin{cases} 1 & \text{if $x\in X$},\\ 0 &\text{otherwise}. \end{cases} $$ At which points ...
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2answers
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If a continuous function on $\mathbb{R}$ $f$ receives an extremum at a single point, it must be the global extremum.

Let $f$ be a continuous function on $\mathbb{R}$ which attains a local maximum at ${{x}_{0}}$. Prove that if $f$ doesn't have any other extremum points, then ${{x}_{0}}$ is the global maximum of $f$ ...
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1answer
32 views

Is this theorem about integration with substitution wrong?

A theorem in my book states: If $g$ is differentiable, f is continuous, and F is an antiderivative of f, then : $\int f[g(x)]g'(x)dx=F[g(x)]+C$ The reason I am asking if this is correct, ...
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1answer
33 views

Functions and continuity proof in real analysis

Prove: If $f\colon A\rightarrow\mathbb{R}^m$ and $a\in A$, show that $\lim_{x\rightarrow a}f(x)=b$ if and only if $\lim_{x\rightarrow a}f^i(x)=b^i$ for $i=1,\dots,m$. The end of the statement is ...
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1answer
89 views
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2answers
57 views

Does Intermediate Value Theorem $\rightarrow $ continuous?

i try to understand Intermediate Value Theorem and wonder if the theorem works for the opposite side. I mean, if we know that $\forall c\:\:\:f\left(a\right)\le \:c\le \:f\left(b\right)\:,\:\exists ...
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2answers
27 views

Checking when an $a$-dependent function is continuous, differentiable.

For some $a\in \Bbb{R}$ define a function $f_{a}(x) = \begin{cases} {x^{a}\cos{1\over x}}, & \text{if $x$ $\ne$ 0} \\[2ex] 0, & \text{if $x=0$} \end{cases}$. Hints firstly are preferred. b. ...
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1answer
11 views

continuity with 2-variables

The question is Determine whether $f$ can be defined at $(0,0)$ so that is is continuous $$f(x,y) = \frac{x^py^q + x^ry^s}{x^qy^p + x^sy^r}, p,q,r,s > 0$$. I chose numbers for p,q,r,s and ...
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1answer
25 views

What variables is $\delta$ dependent on in the epsilon-delta definition of continuity?

The definition of continuity is: $f$ is continuous at $a$ if: Given any $\epsilon>0 $, $\exists \delta > 0$ st. $|x-a|<\delta \implies |f(x)-f(a)|< \epsilon$ $\delta$ obviously depends ...
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1answer
35 views

Which of these statements about a continuous function is true? [on hold]

A function $f$ is continuous on the interval $[0, 2]$. It is known that $f(0) = f(2) = -1$ and $f(1) = 1$. Which one of the following statements must be true? (A) There exists a $y$ in the interval ...
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2answers
39 views

Example of continuous function over $\mathbb R^n$

Let $f:[0,1]\to\mathbb R^n$ such that $f(t)=ty+(1-t)x$ for some $x,y \in \mathbb R^n$. Prove that $f$ is continuous. I know a definition that A function $f\colon X \rightarrow Y$ between two ...
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1answer
28 views

How to tell if a function is continuous at (0,0)

I have to decide if the following function is continuous at (0,0). it's f(x,y) = x^2 + y^2 if (x,y) does not = 0, and f(x,y) = 2 if (x,y) = (0,0) so for the first one, I assume it is continuous, ...
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2answers
25 views

Continuous functions and open sets

I'm working on a proof and having trouble applying a certain theorem. I want to prove that if $ f $ is a continuous function from a metric space into the real numbers, then the set $ {f(x)>0} $ ...
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1answer
31 views

Proof: Cauchy sequences and uniform continuity

I'm working on a proof and I'm having trouble relating definitions I want to prove that if f is uniformly continuous, then if a sequence $ {a_n} $ is Cauchy, $ {f(a_n)} $ is Cauchy. So if $ f $ is ...
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0answers
25 views

Continuous and additive function is linear [duplicate]

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function and $f(x+y)=f(x)+f(y)$, show that $f(x)=kx$, $k\in \mathbb{R}$. I tried to define $g(x)=f(x)-kx$ and $g(0)=0 $ but don't know how to ...
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1answer
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Difficulty with Jensen's Equation.

Its easy to find all continuous function $f: \Bbb R \to \Bbb R $ satisfing the Jensen equation $$f \left( \frac{x+y}{2}\right )=\frac{f(x)+f(y)}{2}$$ But I am finding difficulty in finding all ...
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1answer
639 views

Difficulty in finding a counterexample

I am finding difficulties in finding a counterexample that if $f\colon (0,\infty) \to(0,\infty) $ is uniformly continuous, this implies that $$\lim_{x\to \infty} \frac{f(x+\frac{1}{x})}{f(x)} =1.$$
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1answer
31 views

Riemann Integral Property for Continuous, Monotonic, Non-negative Function

If $f$ is continuous, non-negative, and monotonically increasing function on $[0,∞)$, then prove that $\int^{x}_{0} f(t)dt\leq xf(x)$ $\forall x ≥ 0$ My attempt: Define $F(x)=\int^{x}_{0} f(t)dt$. ...
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1answer
27 views

solve equation with Intermediate value theorem…

set $a_1$,$a_2$,$a_3>0$ and $λ_3>λ_2>λ_1$ on $ℝ$. show that there are exactly two $x$’s for $a_1/(x-λ_1) + a_2/(x-λ_2) + a_3/(x-λ_3) = 0$ I tried use the intermediate value theorem but I ...
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0answers
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The definition of continuously differentiable functions

When we say $f \in C^1$, we mean that f is continuously differentiable. Isn't the continuity a redundant word? I mean, we have a theorem that says if $f$ is differentiable then it is continuous. So ...
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1answer
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three elementary problems on limits of several variable . [on hold]

I'm learning limits of several variable new. Can anyone help me? Computing the following limits: $\lim_{(x,y)\to(0,0)}|x|^y$ $\lim_{(x,y)\to(0,0)}\sin(x/y)$ $\lim_{(x,y)\to(0,0)}x^2\cdot ...
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2answers
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Properties for functions $f:[a,b] \to \mathbb R$? [closed]

Let $f:[a,b] \to \mathbb R$ be a function. Which of the followings are true: A) If $f(x)$ is continuous then it is bounded. B) If $f(x)$ is continuous then it is increasing. C) If ...
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2answers
35 views

Questions about absolutely continuous function

It could be a silly question but the definition of absolutely continuous function says that "A real valued function $f:[a,b]\rightarrow\mathbb{R}$ is absolutely continuous on $[a,b]$ if for all ...
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4answers
43 views

Prove a Continuous Distribution Function is Uniformly Continuous

Let $F$ be the distribution function for a random variable $X$ and it is given that $F$ is continuous over the entire real line. Prove that $F$ is uniformly continuous over the real line. My ...
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1answer
19 views

derivative of a function has only 2nd kind discontinuities

How would I be able to show the following claim If $f$ is differentiable with a finite derivative in an interval, then at all points $f'(t)$ is either continuous or has a discontinuity of the second ...
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1answer
25 views

Smooth saturation function

I need a function similar to $$Saturation(x)=min(max(x, -1), 1)$$ except for I need it to be smooth with no jump in its derivatives. It seems $arctan$ is not a good candidate since I need it to keep ...
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1answer
22 views

Sequential continuity on metric spaces

Please give me a hint for proving this statement: Let $(X,d)$ and $(Y,d')$ be metric spaces, $f$ a function from $X$ to $Y$. If $f^{-1}(B) $ is closed in $X$ for all closed subset $B$ of $Y$, then ...
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2answers
31 views

A sequence of Continuous Functions Converges Uniformly over $\mathbb{R}$ if it Converges Uniformly over $\mathbb{Q}$

I'm trying to show that if ${f_n}$ is a sequence of real functions that is continuous over all of $\mathbb{R}$ and that converges uniformly to $f$ over $\mathbb{Q}$, then it converges uniformly to $f$ ...
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1answer
57 views

For every continuous function $f:[0,1]\to[0,1]$ there exists $y\in [0,1]$ such that $f(y)=y$ [duplicate]

I want to prove that if we have a continuous function from the closed interval [0,1] to the closed interval [0,1], that there exists a value y in [0,1] such that $f(y)=y$. I have an idea of a theorem ...
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1answer
31 views

example of a monotone non-continuous map.

Let me start by defining some terminology to be sure I made no errors there. Parts of this are translated freely from my mother tongue so feel free to correct terminology or the definitions themselves ...
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1answer
53 views

If $f$ is continuous and bounded on $(a,b)$, is it true that $f$ is Riemann-integrable on $[a,b]$? [closed]

If $f$ is continuous and bounded on $(a ,b)$ or $(a, b]$ or $[a, b)$, is it true that $f$ is Riemann-integrable on $[a,b]$? Thanks.
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2answers
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Is this proof that $g$ is continuous correct?

I have proved that $g$ is continuous on $(0,2)$ and I just wish to check if my solution for $g$ being right continuous at $0$ and hence continuous at $0$ is correct. $$\lim\limits_{x \to 0^+}g(x) ...
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1answer
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Continuity of function $f(x) = \lim_{n \rightarrow \infty } (cos(x-\pi/4))^n$

How to check continuity of such function? For me it will be continuous for $x \neq \pi/4+n\pi$ because for such x $|(cos(x-\pi/4))|<1$ But I'm not sure. Can someone tell me if I'm wrong or not? ...
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0answers
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Which one is true? [on hold]

Please give explanation against each option: Which of the following statements are true? a. Let $psi$ be a non-negative and continuously differentiable function on $]0,\infty[$ such that ...
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1answer
31 views

Regarding sup and inf of a continuous function

Suppose $f:\mathbb R\to \mathbb R$ be a continuous function such that $\lim\limits_{x\to \infty}f(x)=0=\lim\limits_{x\to -\infty} f(x)$. Then I want to show that $f$ is bounded and attains at least ...
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1answer
41 views

Determine c and d so that f(x) is continuous if.. [closed]

$$f(x)=\begin{cases}2x^2+cx+d &\text{ for }x<-3\\4&\text{ for }x=-3\\dx^2+2x+c&\text{ for }x>-3\end{cases}$$
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0answers
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How to estimate norms involving $|a-b|$?

I know the title isn't the best one. Here's my problem: Whenever I'm given functionals such as: $$\phi: \mathcal{C}^1([0,1]) \ni f \rightarrow f(\frac{1}{3}) - f'(\frac{2}{3}) \in \mathbb{R}, \ \ ...
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2answers
20 views

Prove this function is lower semi-continuous

Let $X$ be a metric space, and $B$ his borel $\sigma$-algebra. Fix $r>0$ Let $\mu$ be a probability measure on $(X,B)$ and define $f(x)=\mu(B(x,r))$. Show that $f$ is lower semi continuous. What ...
3
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1answer
29 views

$l_2$ sequence, series with square root

I'm trying to prove that the following functional is continuous: $$\phi : \mathcal{l}_2 \ni \{x_n \} \rightarrow \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}x_{3n} - \sum_{n=1}^{\infty} \frac{1}{n}x_{2n} ...
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1answer
20 views

Polynomial, bounded functional

In order to prove continuity of the functional $$\varphi: \mathbb{R}[X] \ni p \rightarrow p'(2011) \in \mathbb{R}$$ where $$||p|| = \sup \{|p(t), \ t\in [0,1]\}$$ I'd like to prove that this ...
2
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2answers
33 views

counter example of equicontinuous

Consider the functions on $[0,1]$: $f_n(x)=nx$, when $x$ is between $0$ and $1/n$ $f_n(x)=2-nx$, when $x$ is between $1/n$ and $2/n$ $f_n(x)=0$, otherwise How to see it is not (uniformly) ...
4
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1answer
54 views

Finding the limit of a sequence of sequences

Take any $\bar{r} \in \mathbb{R}$ with $\bar{r}>0$. Assume that $f : \mathbb{R} \rightarrow \mathbb{R} $ is continuous. Assume that for all $r\in [0,\bar{r})$, there exists a strictly decreasing ...
2
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1answer
18 views

Find the maximum number of a continuous function

Lets define a function $z:\mathbb{R}^\mathbb{R}\to\mathcal P(\mathbb R)$ that gives you the set of zeros of any $\mathbb R ^\mathbb R$ function. Now, we define a set $S=\{z(f):f\in\mathbb R ^\mathbb ...