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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Use the Intermediate Value Theorem to show that $e^x = \sin(x)$ has infinite solutions.

I am not entirely sure about how to prove this algebraically. Although, I can clearly see that $e^x = \sin (x)$ has infinite solutions when I plot the graph. I can't seem to figure out how to ...
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26 views

Checking of continuity

While compactness & connectedness are preserved under continuous maps, this question comes to my mind: $f : \mathbb R \to \mathbb R$ is strictly monotone increasing function such that {$ f(x) : x ...
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38 views

Just a question regarding continuous differentiability

$ f: [0,1] \to [0,1] $ be a MONOTONE & CONTINUOUS function. Does it always imply that: $ f(x) $ is continuously differentiable??
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A question about an epsilon-delta proof

Currently, I am stuck on a question: Let $ g : [ 0 , \infty ) \mapsto \mathbb{R} $ be defined by $g(x)= \left\{ \begin{array}{ll} x^2 & \mbox{if } 0 \leq x \leq 1 \\ 3x & \mbox{if } x ...
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Find the constant $c$ that makes $f$ continuous on $(-\infty,\infty)$

where: $f(x)= x^2-c^2 $ if $ x<4$ $f(x)= cx+20 $ if $ x>=4$ Studying for my midterm: So continuous only means that the two lines line up. So: $4^2-c^2=4c+20$ $4^2-20=c^2+4c$ ...
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How to prove that $x/y$ is continuous in R

$f:R^2$ \{y=0} $\Rightarrow R$ , $f:(x,y)\Rightarrow x/y$. Prove (formally) that $f$ is continuous. I think what I should show is that any point that belongs to an open ball of radius $\epsilon$ of ...
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174 views

How prove this $\lim_{x\to+\infty}(f'(x)+f(x))=l$

let $f(x)$ is continous and $f'(x)$ is continous on $[0,\infty)$,show that $$\lim_{x\to+\infty}(f'(x)+f(x))=l$$ if and only if: $\displaystyle\lim_{x\to+\infty}f(x)=l$ and $f'(x)$ is uniformly ...
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1k views

The definition of locally Lipschitz

I am given this definition: A function $f:A\subset\mathbb R^n\to\mathbb R^m$ is locally Lipschitz if for each $x_0\in A$, there exist constants $M>0$ and $\delta_0 >0$ such that ...
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1answer
69 views

continuous implies frechet differentiable?

I knew if $f$ is Frechet differentiable at $x$ then $f$ is continuous at $x$. But reverse, i.e. If $f$ is continuous at $x$ then $f$ is Frechet differentiable at $x$ true or false?. I think it is ...
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70 views

Is it possible to have a function differentiable but not continuous in a given interval?

Is there any possible function that is not continuous but differentiable in a given interval. It sounds non-logical to me since differentiation is a special limit function in itself therefore ...
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42 views

Conditional probability in continuous distribution

I'm struggling understanding how conditional probability for a continuously distributed random variable is to be calculated. The task is as follows: $f(t) = 1/8 * (4-t)$ for $0 < t <= 4 $ and ...
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1answer
137 views

Extending a $C^1$ function up to the boundary

Let $U \subset \mathbb{R}^N$ be an open bounded set. let $f \colon U \to \mathbb{R}$ be a $C^1$ function. I know that it is not always possible to extend $f$ continuously up to the boundary of $U$ ...
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133 views

Proving continuity and differentiability of a function

let $f(x)=\sin |x|, x\in (-\pi,\pi)$. Is $f$ continuous on $(-\pi,\pi)$ ? Is it differentiable in that interval ? I have read continuity at a point. How do i prove it for an interval ?
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32 views

A basic doubt on showing some function is uniformly continuous

I want to prove that the function $f(x)=x^2$ is uniformly continuous in the interval $[0,1]$. For that I am taking two sequences $x_n$ and $y_n$ in $[0,1]$ such that $x_n - y_n \to 0$, then I want to ...
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1answer
41 views

About a function continuous on [0,1]

Let $f:[0,1]\to [0,1]$ be a continuous function. Given that $f(0)=0$, $f(1)=1$ and $f(f(x))=x$ prove that $f(x)=x$. I've been thinking about this one for ages, but I can't figure out even where to ...
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1answer
50 views

Moving boundaries for Ornstein-Uhlenbeck processes

Let $\tau(X_t)$ be the first-passing time to the moving boundary $a(t)$ for an Ornstein-Uhlenbeck process $X_t$. I wonder how general an $a$ can be allowed in order to guarantee that $\tau$ becomes ...
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89 views

A real analysis problem on continuous functions

Let $f$ be a continuous mapping from closed interval $[0,1]$ to itself. I need to prove that $f(x)=x$ for at least one $x \in [0,1]$. I want to do it by contradiction i.e. assuming that $f(x) \neq x ...
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2answers
44 views

A property of continuous functions [duplicate]

A function $f : [a,b] \to [a,b]$ is continuous for all $x \in [a,b].$ Prove that there exists a $c\in [a,b]$ such that $f(c) = c.$
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49 views

Proving that half an isometry is a homeomorphism

Let $(K,d)$ be a compact metric space and $f:K\rightarrow K$ such that $$\forall x \in K, \forall y \in K, d(f(x),f(y)) \geq d(x,y)$$ Prove that $f$ is a homeomorphism. What I managed to prove is ...
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130 views

Maps with every point being periodic

Does there exists a characterization of continuous maps $f:[0,1]\rightarrow [0,1]$ with every point $x\in [0,1]$ being periodic (i.e. if for every $x\in [0,1]$ there exists $n\in\mathbb{N}$ such that ...
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256 views

What's the importance of continuous functions and continuity?

While studying calculus, I've read about continuous functions but I still couldn't figure out what's the importance of the concept, I imagine that the concept (and also the concept of continuity) may ...
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Showing continuity of partially defined map

There is a theorem in Note on Cofibrations by Arne Strøm. It says Let $A$ be a closed subspace of a topological space $X$. Then $(X,A)$ has the HEP if and only if there are (i) a neighborhood ...
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188 views

Examine the continuity and discontinuity of the following function at $(0,0)$

Examine the continuity and discontinuity of the following function at $(0,0)$ $$f(x,y)=\begin{cases}{x^3\cos({1\over y})+y^3\sin({1\over x})\over {x^2+y^2}} & x\neq 0\neq y\\ 0 & ...
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Prove $f(x) = 0 $for all $x \in [0, \infty)$ when $|f'(x)| \leq |f(x)|$

mathematicians! I want to ask to all wise people about a problem I met at the quiz to obtain some ideas. The problem is following. It may not be accurate since the problem is dependent on my memory ...
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1answer
55 views

Hausdorff spaces from continuous functions

The question is to prove a topological space is Hausdorff if for every $p$ in the space there exists a continuous function $f_{p}$ such that $f^{-1}(0) = \{p\}.$ (The inverse here is implied as ...
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1answer
38 views

What can be said about continuity and differentiability of $f$ on $\mathbb{R}$?

$f(x)=\lim_{n\to \infty}(\sin^{2n} x+\cos^{2n} x)^{1\over 2n}$ What can be said about continuity and differentiability of $f$ on $\mathbb{R}$? $f(x)=1$ when $x=2k\pi$ and also multiple of ...
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35 views

closed maps on topological spaces

Prove that $f$ is closed if and only if $f(\text{cl}(A)) \supseteq \text{cl}(f(A)).$ I can show the forward directions by saying: Suppose $f$ is closed. Then, $f(\text{cl}(A)) = f(A) \cup f(\partial ...
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Limits (discontinuity)

Determine the points where the function with domain R is not continuous. Identify as removable or an essential discontinuity. $$ f(x)= \begin{cases} &\frac{x^2-1}{x-1} \text{ if } x\neq 1 \\ ...
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196 views

How to prove that $f(x,y)=\sqrt{x^2+y^2}$ is continuous in $\mathbb{R}^2$? [duplicate]

Please, I need the demonstration (step by step) of the continuity in $\mathbb{R}^2$ of the function $f(x,y)=\sqrt{x^2+y^2}$. I know that the function is continuous in $\mathbb{R}^2$, but I just don´t ...
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1answer
375 views

Showing a function is Hölder Continuous

I'm trying to show that the function $f(x) = x \ln(x)$ is Hölder continuous on $(0,1) $ for $0 < \alpha < 1$. I must be missing something, because I don't really understand how the definition ...
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If $f^{-1}(G)$ is open in $X$ for every open set $G$ in $Y$, then $f$ is continuous. Question on proof.

Let $X,Y$ be metric spaces and $f:X\rightarrow Y$. If $f^{-1}(G)$ is open in $X$ for every open set $G$ in $Y$, then $f$ is continuous. The text I am using proves this proposition like so: Suppose ...
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1answer
205 views

Show convergence of a sequence of continuous functions $f_n$ to a continuous function $f$ does not imply convergence of corresponding integrals.

Let $f_n\in C([0,1])$ be a sequence of functions converging uniformly to a function $f$. Show that $$\lim_{n\rightarrow\infty}\int_0^1f_n(x)dx = \int_0^1 f(x)dx.$$ Give a counterexample to show that ...
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101 views

Continuity for topological spaces

After reading the definition of a continuous map on general topological spaces, my question is the following: Suppose $f$ is continuous from $\mathbb R$ to $\mathbb C$ given by $x \mapsto e^{ix}$. ...
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1answer
72 views

$F = \{f\in C^1([0,1])| \hspace{2mm} \|f\|\leq M, \|f'\|\leq N\}$. Showing it is precompact and not closed.

I have an example in my book: Let $C([0,1])$ denote the space of all continuous functions $f$ on $[0.1]$ with continuous derivative $f'$. For constants $M>0$ and $N>0$, we define the subset $F$ ...
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Prove that $C^1([a,b])$ with the $C^1$- norm is a Banach Space

Consider the space of continuously differentiable functions, $$C^1([a,b]) = \{f:[a,b]\rightarrow \mathbb{R}|f,f' \text{are continuous}\}$$ with the $C^1$-norm $$||f|| = \sup_{a\leq x\leq ...
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1answer
206 views

Is $f$ continuous at $(0,0)$

$$ f(x,y) = \begin{cases} \frac{xy^2}{x^2 + y^2} & \text{ if } (x,y) \neq (0,0) \\ 0 & \text{ if } (x,y) = (0,0)\end{cases} $$ (i) Is $f$ continuous at $(0,0)$? At $(x,y) \neq (0,0)$ this ...
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1answer
217 views

Let $X$ be a compact metric space. If $f:X\rightarrow \mathbb{R}$ is lower semi-continuous, then $f$ is bounded from below and attains its infimum.

Let $X$ be a compact metric space. If $f:X\rightarrow \mathbb{R}$ is lower semi-continuous, then $f$ is bounded from below and attains its infimum. I want to prove this. This is my proof: Since $X$ ...
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1answer
68 views

Prove/disprove statements regarding continuous functions.

I found some old math tests from my school years and thought it might be fun to see what I still remember. The answer is simply, not as much as I hoped for. I'm having trouble proving/disproving these ...
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Making a multivariable Function continuous

This function $$f(x,y)=\frac{e^{xy}-\cos (x)+\sin(xy)}{x}$$ can be made continous for $f(0,y)$ by defining $$f(0, y) = 2y .$$ My question is: how can i get to this conclusion ("$2y$ must be it") ...
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130 views

Determine the set of points where $f$ is continuous.

Define $f:[0,1]\rightarrow\mathbb{R}$ by $$ f(x) = \begin{cases} x & \text{if $x$ is irrational} \\ p\sin(\frac{1}{q}) & \text{if x=$\frac{p}{q}$, where $p,q$ are relatively prime integers.} ...
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126 views

Does extreme value theorem hold when continuous is replaced with bounded?

The extreme value theorem says that if the domain of a 'continuous' function is compact then both the max and min of the function lies in the domain set. My question is: can the 'continuity' be ...
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1answer
147 views

Epsilon-delta proof of the existence of the limit of a sequence?

If $\lim_{n\to\infty}a_n \rightarrow L$ and the function $f$ is continuous at $L$, then $$\lim_{n\to\infty}f(a_n) \rightarrow f(L)$$ $\underline{Proof.}$ Let $n, N \in \mathbb{N}$. Let ...
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96 views

A function continuous in both arguments

Is there a two-arguments function which is not continuous but continuous in each argument? It seems I have studied something like this, but don't remember.
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63 views

Smoothly extending a smooth function.

Ok so maybe it is just late but I am embarrassed to say that I am a little stumped by this question which is noted by the professor to just be a "calculus exercise". Anyway, let $f$ and $g$ be smooth ...
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33 views

Dependency of differential entropy on values of random variable

For a discrete random variable, entropy does not depend on the values-, but only probabilities of that random variable. Does this property also scale to the continuous case, when random variable X ...
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128 views

Why is $\lim_{x \to c}g(f(x)) = g(\lim_{x \to c}f(x))$

In this theorem (from the continuity section of the first chapter of a calculus textbook) If $g$ is continuous at $b$, and $\lim_{x \to c}f(x)=b$, then $\lim_{x\to c}g(f(x)) = g(\lim_{x \to ...
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127 views

Understanding a periodic discontinuous function

This is from an example in my PDE text. It's something I should probably know, but maybe I am just reading the wording wrong. The text (Asmar as it happens, section 2.1) gives the following example ...
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43 views

Is this any measurable function that the set B is the collection of points that makes f continuous for any Borel set B?

I can find that the function f \begin{align} f(x)= & 1/p & (x=p/q, p/q ~ simple) \\ { } & 0 & (x ~ is ~ irrational) \end{align} is continuous only on x for x irrational and ...
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1answer
35 views

Help me understand a consequence of continuity.

Let $f:[-\pi, \pi] \rightarrow \mathbb{C}$ be a Riemann-integrable function that is continuous at zero. Since $f$ is continuous at $0$, we can choose $0 \lt \delta \leq \pi/2$, so that $f(\theta) ...
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323 views

Why is $\sin(xy)/y$ continuous?

Me and my mates are crunching this question for a while now. While we know that $\sin(xy)$ is continuous , $1 / y $ as the other part of the function clearly has a continuity gap at $y = 0 $, though ...