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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Provide an example of each or explain why the request is impossible.

(a) Two functions $f(x)$ and $g(x)$, neither of which are continuous at $0$ but $f(x)+g(x)$ and $f(x)g(x)$ are both continuous at $0$ I said possible and let $f(x) = \{0: x<0, 1: x \geq 0 \}$, ...
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39 views

I do not understand the last process of proving that $f$ is continuous iff $f^{-1}(G)$ is open.

The problem is: Let $f$ be a finite function on $\mathbb{R}^n$. show that $f$ is continuous on $\mathbb{R}^n$ if and only if $f^{-1}(G)$ is open for every open $G$ in $\mathbb{R}^1$, or if and ...
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What is the logic underlying this proof?

Proposition: A metric space $X$ is connected if, and only if, every continuous function $f:X\to (\{0,1\},d_D)$ is a constant function, where $d_D$ is the discrete metric on the set $\{0,1\}$. ...
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28 views

Prove that if $x\mapsto -x$ is continuous then $\sigma$ is the discrete topology.

Let $\tau $ be the topology on $\Bbb R$ for which the intervals $[a,b)$ form a base.Let $\sigma$ be a topology on $\Bbb R$ such that $\sigma \supseteq \tau. $ Prove that if $x\mapsto -x$ is ...
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23 views

show that continuous functions on $\mathbb{R}$ are measurable

I am trying to show this using the theorem: A function $f: \Omega \to \mathbb{R}$ is measurable if and only if $f^{-1}(E) \in \mathcal{F}$ for all borel sets $E$. The proof to show a continuous ...
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1answer
42 views

Are $\lim_{h\to0}f(a+h)=f(a)$ and $\lim_{h\to0}f(x+h)=f(x)$ the same?

An exercise I came across in my calculus text is as follows: Prove that $f$ is continuous at $a$ if and only if $$\lim_{h\to0}f(a+h)=f(a)\tag{1}.$$ Now, I saw a proof of the Product Rule ...
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2answers
25 views

Checking if “continuous” when $x$ is 1 and reaches 1

I have $$f(x) = x \left| x - 1 \right|$$ Here my given value for $x$ is 1 And I need to test if the function is "continuous" when $x$ is $1$ and also when reaching $$ f(1)$$ $$ \lim\limits_{x \to ...
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31 views

When $F(t)=\int_0^tf(s)ds$ is differentiable everywhere?

Let $f:\mathbb{R}\to \mathbb{R}$ be a function that is continuous almost everywhere. 1) Is the function $F(t)=\int_0^tf(s)ds$ differentiable everywhere ? 2) What is the "weakest" condition on $f$ ...
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45 views

Function that is second differential continuous

Let $f:[0,1]\rightarrow\mathbb{R}$ be a function whose second derivative $f''(x)$ is continuous on $[0,1]$. Suppose that f(0)=f(1)=0 and that $|f''(x)|<1$ for any $x\in [0,1]$. Then ...
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Continuous function rational for every point, Cantor function

For Cantor function (https://en.wikipedia.org/wiki/Cantor_function), in my sense it is rational on every point. But it is continuous on [0,1], then such a function must be constant. What is the ...
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25 views

Let $\alpha$ be a real number. Find the value of $\alpha$ for which the given function is continuous and differentiable.

Let $\alpha$ be a real number. Consider the function $$g(x)=(\alpha+|x|)^2e^{(5-|x|)^2}, \ \ \ -\infty<x<\infty $$ $(i)$ Determine the values of $\alpha$ for which $g$ is continuous at all $x$. ...
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2answers
44 views

Using the $\epsilon-\delta$ definition show that $f(x) = \frac 1 {x^2}$ is a continuous function at any $x_0 = a, a > 0$

Using the $\epsilon-\delta$ definition show that $f(x) = \frac 1 {x^2}$ is a continuous function at any $x_0 = a, a > 0$ I have expressed in the form: $$lim_{x\to a}\frac1{x^2}=\frac1{a^2}$$ ...
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5answers
20 views

Prove the continuity on an open interval

I need to show, that function $f(x) =\frac{2x +3}{x-2}$ is continuous on the interval $(2,\infty)$ My attempt: We should find the right-hand limit to prove the continuity: and this limit is equal to ...
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25 views

If f*g is Riemann integrable, g continuous, nonzero and bounded, show that f is Riemann integrable

How would I go about proving that if $fg$ is Riemann integrable, given that $g$ is continuous, nonzero, and bounded (so $g$ Riemann integrable), how would I go about showing that $f$ is Riemann ...
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32 views

Function is continuous if graph is compact.

Let $X$ be a Hausdorff space and let $f:X\to \mathbb{R}$. If grapph of $f$ is compact we have to show that $f$ is continuous. Since every closed subset of a Hausdorff space is closed, therefore ...
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2answers
56 views

Using the $\epsilon$-$\delta$ definition, show that $f(x) = \frac 1 {x^2}$ is a continuous function at any $x_0 = a, a > 0$

Using the $\epsilon$-$\delta$ definition show that $f(x) = \frac 1 {x^2}$ is a continuous function at any $x_0 = a, a > 0$. To what I understand of this question, is it just asking to me ...
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2answers
17 views

let $f$ be a function defined on all of $\mathbb{R}$

Is there a function that is continuous in this specific manner? $\forall \epsilon >0$ we can choose $\delta = \epsilon$, and it follows that $|f(x) - f(c)| < \epsilon$ whenever $ |x-c| < ...
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32 views

Continuity, algebraic and rational numbers [on hold]

Is it true that there exist a continuous function f that for every algebraic number q , his image f(q) is a rational number? Thank you for your answers
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Comparing smoothness among approximations

We are interpolating a missing fragment of a 2D curve given a set of sample points. Our method generates several candidates of curve pieces to fill the missing part, but we want to select the solution ...
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3answers
45 views

Rigorous Definition of One-Sided Limits

In a typical first-year Calculus course professors typically tend to put a lot of emphasis on making visual connections when working with "one-sided" limits or derivatives. This is something I find ...
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1answer
34 views

Prove $\{f_n(x)\}$ is not continous

$f_n(x) = \left\{ \begin{array}{ll} \frac{1}{n} & \quad x \in \mathbb{Q} \\ 0 & \quad x \notin \mathbb{Q} \end{array} \right.$ Not sure how to show ...
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1answer
41 views

How can one prove that a real function is closed? [on hold]

I am defining a closed function to be one that takes closed sets to closed sets. Given a function, domain and codomain, you could prove that it is not closed by simply providing a counter example ...
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39 views

Preimage of sets, complement of sets, continuity of functions

I just got some simple questions in real analysis regarding preimage and complement of sets and continuity. Suppose $f:X\to Y$, then does $f^{-1} (Y\setminus F)=f^{-1} (Y)\setminus f^{-1} ...
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46 views

Example of a function that converges to 0 pointwise but integral is 3/2?

Give an example of a sequence of continuous functions $(f_n)$, $f_n : [0, 1] \to \mathbb{R}$ that converges to zero pointwise, and such that the integral of each function within the given domain is ...
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3answers
143 views

Uses of step functions

My highschool teacher has informally told us about what continuity is and used step functions as an example of a discontinuous function. The Wikipedia page for it links to a lot of other kind of step ...
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33 views

Let $f$ be a real-valued continuous function on $[0,1]$ which is twice continu-ously differentiable on $(0,1)$. Suppose that $f(0) = f(1) = 0$

Let $f$ be a real-valued continuous function on $[0,1]$ which is twice continu-ously differentiable on $(0,1)$. Suppose that $f(0) = f(1) = 0$ and $f$ satisfies the following equation: $$x^2f''(x) + ...
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1answer
33 views

Continuous function on a non-compact set

I'm trying to show if $X$ is non compact ($X \subseteq \mathbb{R}$) then there is a cont function $f:X \rightarrow \mathbb{R}$ which is bounded but doesn't attain it's bounds. I'm trying it for a set ...
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1answer
32 views

Explain how L(g,P) = U(g,P) implies continuity of g.

First, let $g$ be bounded on $[a,b]$. Now, assume $\exists P$, a partition, such that $L(g,P)=U(g,P)$. I am told the correct answer to the question "describe $g$" is that $g$ is continuous on ...
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1answer
25 views

Function of metric with a fixed point

I'm trying to prove that given a metric space $(X, d)$, for a fixed $x\in X$, define the function $g(y)=d(x,y)$, then $g(y)$ is continuous, using triangle inequality. My first question is that can I ...
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1answer
38 views

What is an example of a continuous but not closed function? [duplicate]

I have two questions about closed functions. Firstly, we say that a function is closed if it maps closed subsets in the domain to closed subsets in the co-domain. Polynomials are typical examples of ...
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24 views

continuity of a piece wise function defined partially on a closed interval

using epsilon delta definition prove that $f(x)=\left \{ \begin{array}{cc} 2 & : x \in[0,1]\\ 1 & : x=-1 \end{array}\right.$ is continuous on $E= [0,1] \cup \{-1\}$. Here is my attempt. I ...
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1answer
28 views

A smooth function which is nowhere real analytic, and preserves rationality of its argument

There are examples $\!^{[1]}$$\!^{[2]}$ of continuous infinitely differentiable (class $C^\infty$) functions $\mathbb R\to\mathbb R$ that are nowhere real analytic. I wonder if it is possible to ...
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1answer
16 views

Is $f(x,y) = (x^2-y^2,xy)$ lipschitz on $\mathbb{R}^2$?

How can I show that the $f$ is lipschitz? I try to calculation such that $|f(x_0, y_0) - f(x_1,y_1)|^2 = ((x_0^2 -y_0^2)-(x_1^2 -y_1^2))^2 +(x_0 y_0 -x_1y_1)^2$ and $|(x_0-x_1,y_0-y_1)|^2 = ...
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13 views

Volume Zero of Not Continuous Function

Show that a bounded real-valued function f on a closed interval $I$ of $E^n$ is integrable on $I$ if and only if the set of points of $I$ at which $f$ is not continuous is the union of a sequence of ...
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22 views

Continuity of a map in a metric space

Let $C^0([a,b])$ denote the space of continuous function $f:[a,b]→\Bbb R$. Define $ d(f,g)= \sup_{[a,b]}|f-g| $. We define $F:C^0([a,b])→\Bbb R$ to be $F(f)=\int_a^b f$. I want to show that $F$ is a ...
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27 views

Is there a way to calculate RMS value continuously?

Using that the RMS by definition is: $\sqrt {\int_0^T\frac 1T*f(t)^2dt} $ which can be calculated by using Riemann sums in the following way: $\sqrt {\frac 1N\sum_0^Nf[i]^2} $ I've tried that in ...
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1answer
25 views

Mean value theorem, Wierstrass theorems

I have a question that is related to these theorems I tried to tackle but got stuck Please let me know if it is the proper way to go ? The question is: Let $f:[0,1]\rightarrow\mathbb{R}$ be a ...
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46 views

Metric space and continuity

We define a map $f:(S,d)→(S',d')$ between 2 metric spaces to be continuous at x belongs to S if for every sequence ${x_n}$ in $S$ that converges to x, the sequence {f(x_n)} in $S'$ is convergent to ...
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Showing a particular type of continuous function is uniformly bounded

Let $I = [0,\infty)$ and $f:I \to I$ be continuous with f(0) = 0. Show that if \begin{equation} f(t) \leq 1 + \frac{1}{10}f(t)^2, \text{ for all } t \in I \end{equation} then $f$ is uniformly bounded ...
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1answer
26 views

Proof of Interceting Lines

I have this practice problem from a final exam study guide. Let $f,g$ be continuous on $[a,b]$ and $f(a)>g(a)$ but $g(b)>f(b)$. Prove that $\exists c \in [a,b]$ such that $f(c)=g(c)$. My ...
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1answer
38 views

Complex supremum function is strictly monotone

I'm having great troubles to solve the following exercise: Let $f$ be a holomorphic function on the unit disc. For $0\leq r < 1$ is $$M(r):=\sup\limits_{|z|=r}|f(z)|$$ Show that ...
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1answer
28 views

Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C_b[0,1]$

Following Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$ I would like to prove that the same is true for bounded functions on $[0,1]$ ...
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Every open ball of a normed vector space $E$ its homeomorph to the entire space $E$.

I have some questions about this proof that "Every open ball of a normed vector space $E$ its homeomorph to the entire space $E$.": By the example (12), we just have to consider the ball $B(0,1)$, we ...
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How is $d(af(x), af(x_o))$ and $d(f(x), f(x_o))$ related?

I wish to prove that given $f \in C_0([0,1])$ of continuous function, then $af \in C_0([0,1])$ where $a \in \mathbb{R}$ I am having trouble relating $d(af(x), af(x_o))$ with $d(f(x), f(x_o))$ So to ...
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1answer
54 views

Prove that $f(x)=\sum_{n=1}^{\infty} (\frac{x}{n}-log(1+\frac{x}{n}))$ is continuous and can be differentiated ad infinitum

We have $f:(0,\infty) \rightarrow \mathbb{R}$ defined by infinite series $f(x)=\sum_{n=1}^{\infty} (\frac{x}{n}-log(1+\frac{x}{n}))$ Prove that $f$ is continuous and can be differentiated ...
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42 views

derivative of differentiable function [duplicate]

Edited: It is known that if $f$ is differentiable then the derivative function of $f$ is not always continuous. For instance $f(x)=x^2\sin (\frac{1}{x})$ for $x\neq 0$ and $f(0)=0$ if $x=0$. Then ...
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1answer
37 views

$f \in C^1[0,\infty)$ such that $\lim_{x \to \infty} \dfrac {xf(x)}{f'(x)}=2$ ; then for $s<2$ ; $\lim_{x \to \infty}x^{-s}f(x)=\infty$?

Let $f \in C^1[0,\infty)$ be such that $\lim_{x \to \infty} \dfrac {xf(x)}{f'(x)}=2$ ; then is it true that for $s<2$ , $x^{-s}f(x) \to \infty$ as $x \to \infty$ ?
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6 views

How can I write a gradient of sobel filter in continuous formula?

Let $*$ denote a convolution operation, $G$ denote a kernel, and $I$ is a given image. The gradient of the image $I$ is equivalent: $\nabla (G*I) = (\nabla G) * I$ The Sobel filter approximtes two ...
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1answer
23 views

The continuity of injectivity radius

Let $M$ be a Riemannian manifold. $r:M\to [0,+\infty]$ denotes the function assigns to $p\in M$ the injectivity radius $r_p$ of the exponential map $\exp_p$. Is this function $r$ is continuous or ...
3
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45 views

Why is continuity permissible at endpoints but not differentiability?

Differentiable at endpoints? cause of differentiation only on an open set. Admittedly, there are some questions and answers as to why a function defined on a closed interval is not differentiable on ...