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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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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1answer
17 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 ...
3
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1answer
39 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 ...
2
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1answer
41 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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4answers
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An inflection point where the second derivative doesn't exist?

A point $x=c$ is an inflection point if the function is continuous at that point and the concavity of the graph changes at that point. And a list of possible inflection points will be those points ...
2
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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 ...
2
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2answers
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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0answers
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 $f$ ...
7
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2answers
414 views

If every real valued continuous function on $X$ is uniformly continuous , then is every continuous function to any metric space uniformly continuous?

Let $X$ be a metric space such that every continuous function $f:X \to \mathbb R$ is uniformly continuous ( here $\mathbb R$ is equipped with the standard euclidean metric ) , then is it true that for ...
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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: $$lim_{x\to a}\frac1{x^2}=\frac1{a^2}$$ ...
2
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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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0answers
16 views

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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0answers
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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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 ...
0
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2answers
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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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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2answers
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
3
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1answer
27 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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2answers
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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0answers
10 views

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 ...
2
votes
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 ...
3
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2answers
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Prove that $f=1/\sqrt{x}$ is continuous on the interval $(0,1]$, but not uniformly continuous.

Prove that $f(x)=1/\sqrt{x}$ is continuous on the interval $(0,1]$, but not uniformly continuous. I believe it follows that $f(x)$ is not uniformly continuous because $f(x)$ is not continuous on the ...
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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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votes
2answers
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 ...
0
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1answer
31 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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0answers
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) + ...
1
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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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votes
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 ...
2
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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 ...
3
votes
2answers
97 views

is the inverse of a absolutely continuous function with almost everywhere positive derivation absolutely continuous?

suppose $f$ is an absolutely continuous on $[0,1]$,that almost everywhere $f'>0$. is the inverse of $f$ necessarily absolutely continuous on $[f(0),f(1)]$? thank you very much!
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0answers
23 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 ...
0
votes
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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0answers
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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2answers
21 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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0answers
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 ...
1
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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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0answers
102 views

onto and strictly increasing imply homeomorphism [closed]

I have this question: Let $f:[\alpha,\beta[\rightarrow [a,b[$ be an onto map and strictly increasing. How to prove that $f$ is a homeomorphism? (This means $f$ and $f^{-1}$ are continuous.) ...
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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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2answers
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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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2answers
83 views

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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2answers
147 views

How to prove that there does not exist a sequence of continuous functions that converge pointwise to $\chi_{\mathbb{Q}}$ (definition only)

A fellow member of the community asked: "there isn't a sequence of continuous function on $[0,1]$ that converges pointwise to the function $f$ on $[0,1]$ defined by $f(x)=0$ if $x$ is rational and ...
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1answer
27 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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1answer
35 views

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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0answers
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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
24 views

Example of convergent sequence and discontinuous function

I need a counter example to show the following statement is false: A function $f$ is continuous at a point c if there exists a sequence $x_n$ such that $x_n \rightarrow c$ as $n \rightarrow \infty$ ...
0
votes
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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1answer
53 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 ...
0
votes
2answers
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 ...