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) ...

learn more… | top users | synonyms (1)

0
votes
1answer
25 views

Continuity in closed sets

Please help me, I have being trying this for days now. Let $f:F \to \mathbb{R}$ be a function on a closed set $F$. Show that $f$ is continuous if and only if $A=\{x \in F; f (x) \leq c\}$ and $B=\{x ...
2
votes
1answer
46 views

Non-continous topology?

I've been studying topology this term and it really got me interested. But sometimes in math I feel that we are just taught things one by one, without really talking about why we do it that way. So I ...
4
votes
1answer
27 views

Trying to prove that a function got no limit at $(0,0)$

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$, defined by: $$ f(x,y)=\begin{cases} 1 & y=x^{2}\\ 0 & \text{otherwise} \end{cases} $$ How can I show that this function got no limit at $(0,0)$? ...
1
vote
3answers
48 views

The derivative of $x!$ and its continuity

is the factorial of fractions and negative numbers defined? If yes, then what is its graph? Also please find its domain. Our teacher said the factorial of a fraction is the fraction itself. He also ...
1
vote
1answer
16 views

Proving a corollary of a corollary of the Mean Value Theorem (corollary-ception)

This is will a wordy question but here it goes: My analysis book states the mean-value theorem and then a corollary which we will label as (1): Let $f$ be a differentiable function on $(a,b)$ such ...
0
votes
2answers
26 views

How to evaluate limits

Let $f$ be a continuously differentiable function on $\mathbb R$. Suppose that $L=\lim\limits_{x\to \infty}(f(x)+f^{'}(x))$ exists. If $0<L<\infty$, and if $\lim\limits_{x\to \infty} f^{'}(x)$ ...
0
votes
2answers
38 views

Is $t\mapsto 1_{[0,t]}(s)$ for a fixed $s\ge 0$ continous?

Let $s\ge 0$ and $$f:[0,\infty)\to\left\{0,1\right\}\;,\;\;\;t\mapsto 1_{[0,t]}(s)$$ Is $f$ continuous at $t_0\ge 0$? If $s>t_0$, then $f(t_0)=0=\displaystyle\lim_{n\to\infty}f(t_n)$ for all ...
3
votes
3answers
49 views

Prove that an increasing and surjective function is continuous.

If $f:[a,b]\rightarrow [f(a),f(b)]$ is increasing and surjective, prove that it is continuous. Fix $c \in (a,b)$. Take $\epsilon >0$. We then wish to find the set of $x$ such that ...
1
vote
2answers
56 views

Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq f(x_2)$ and let $c \in (a,b)$. Show that limits from both sides exist.

I would like to ask you a question about the following question. Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq f(x_2)$ and let $c \in (a,b)$. Show that $\lim_{x \ \rightarrow ...
2
votes
1answer
37 views

Using lipschitz estimate to show $|f_n(x) - f_p(x) - (f_n(c)-f_p(c))| \leq |b-a|\sup_{y \in (a,b)}|f'_n(y)-f_p'(y)|$

Assume $(f_n)$ is a sequence of functions that are continuous on $[a,b]$ and differentiable on $(a,b)$. Then using Lipschitz estimate to prove that $$|f_n(x) - f_p(x) - (f_n(c)-f_p(c))| \leq ...
0
votes
1answer
18 views

Extending a function continuously from a subset to the whole set

We are given two sets $E$ and $F$ such that $F \subset E \subset \mathbb{R}$. We are given a continuous function $f$ defined on $F$. Can we always extend it to a continuous function on E (not ...
3
votes
1answer
51 views

Show $\sum e^{-nx + \cos(nx)}$ is defined on $(a, \infty) $ for any $a>0 \dots$

I want to prove that $\sum e^{-nx + \cos(nx)}$ is defined and continuous on the given interval of $(a, \infty)$ where $a >0$. Then, how exactly do I show it is defined? It just seems trivial, ...
0
votes
1answer
30 views

Simultaneous density function of two continuous variables, X and Y.

I'm having issues with calculating the simultaneous density function of two continuous variables, X and Y. I took a screenshot of the information: How should I start? I know that if the two ...
1
vote
0answers
47 views

How to Find the pointwise limit of $(f_n)$

For $x \in [0, \pi/2]$, if $$f_n(x) = \frac {nx} {1+n\sin(x)}$$ how do you find the pointwise limit of $(f_n)$ ?
1
vote
1answer
20 views

Continuity of multivariable functions

I have a question regarding norms on $\Bbb R^{n}$ and proving the continuity of multivariable functions. Specifically, suppose we have $f: \Bbb R^{2} \to \Bbb R$, for example. To prove $f$ is ...
3
votes
2answers
56 views

Proving that a function is discontinuous

In my assignment I have to prove that the following function is discontinuous: $$f(x)=\begin{cases}2x-1&\text{if }x\notin\Bbb Q\\x^2&\text{if }x \in \Bbb Q\end{cases}$$ I have to prove that ...
0
votes
2answers
21 views

A question involving continuity with respect to the product topology

Let H be a nonempty set, $\cdot$ a binary operation on H, $\Gamma$ a topology on H and $$\varphi : H \times H \to H, \;\; \varphi(x, y) = x y, \;\; \forall x, y \in H$$ continuous with respect to the ...
-1
votes
1answer
24 views

Check differentiablity of $f$ [closed]

Consider a function \begin{equation*} f(x)=|\cos x|+|\sin (2-x)|. \end{equation*} At which of the following points f is not differentiable? a)$\{(2n+1)\frac\pi2:n\in \Bbb Z\}$ b)$\{n\pi:n\in \Bbb ...
-1
votes
0answers
24 views

Idempotent ideal in ring of continuous functions

Is there any equivalence conditions under which an ideal $I$ in ring of continuous functions be an idempotent ideal?
-2
votes
0answers
24 views

A “prove or disprove question” on absolutely continuous functions [on hold]

Let $f:\left[a,b\right]\rightarrow\mathbb{R}_{+}$ be an absolutely continuous function. ($a<b$). Prove or disprove that the right ( respectively left) derivative of f exists at each point of the ...
-1
votes
2answers
20 views

Sequential Definition of continuity || Modulus Property

I am stuck up with these questions from my text book on sequential continuity : {My questions might sound trivial a bit trivial} I am not able to figure how its being written that $|f(X_n)| \leq ...
1
vote
1answer
25 views

Is the function $f(x)=x^2$ absolutely continuous on the real line?

In Wiki (Lipschitz), it says: A Lipschitz function $g : \mathbb{R}\to \mathbb{R}$ is absolutely continuous. According to the definition of absolute continuity, I am confused about an simple ...
0
votes
2answers
28 views

Continuous and bounded - Check my proof please

Let $f : [0, ∞) → \mathbb{R}$ be continuous such that $\lim_{x→+∞} f(x) = 0$. Prove that $f$ is bounded on $[0, ∞)$ By our hypothesis and the definition of continuity, given $ c \in [0, \infty), ...
1
vote
1answer
36 views

Prove that $f$ in monotonic

In my assignment I have to prove the following: Let $f$ a continuous function in $\Bbb R$. Prove the following: if $|f|$ is monotonic increasing, in R then $f$ is monotonic in R. ...
0
votes
2answers
30 views

Solving $f(x) = x^5 +x + 1 = 0$ with halving the interval / bisection method

Question: Use halving the interval / bisection method to approximately solve: $$f(x) = x^5+ x + 1 = 0$$ with a precision of $\pm 0.1$ Attempted solution: The general idea, as I understand it, is ...
1
vote
2answers
25 views

Extreme value theorem, without Heine Borel.

I was wondering, if there are any mistakes, in this proof of the extreme value theorem: Theorem. Let $X$ be a compact set and $f:X\rightarrow\mathbb{R}$, s.t. $f$ is continuous. Then there exists ...
0
votes
2answers
51 views

Is the function continuous - $f(x) = \frac{1}{\sin x} + \frac{1}{x-1}$

I have an assignment in which I have to prove that a function "recieves every real value, where $x\in (0,1)$". Here is the function: $$f(x) = \frac{1}{\sin x} + \frac{1}{x-1}$$ I don't know the ...
1
vote
1answer
35 views

Can a function be continuous at the end points of its (closed interval) domain?

Assume $f$ has a domain of $[a, b]$. Is it possible that $f$ is continuous at $x = a$ and $x = b$? If the definition of continuity is that the left and right limits are equal to the function at the ...
3
votes
2answers
68 views

When is the function Continuous?

In my assignment I have to determine when is the function continuous. This is the function: \begin{equation} g(x) = \begin{cases} \left\lfloor {\sin\frac{1}{x}}\right\rfloor&\text{if} \space ...
1
vote
1answer
28 views

Limit vs interior definition of continuity

Suppose I have two topological spaces $X$ and $Y$ whose topologies are defined by interior operators $\text{int}_X$ and $\text{int}_Y$ respectively, as well as a function $f$ with domain $I$ (for ...
1
vote
2answers
64 views

Is this condition on continuity extraneous or troublesome?

I was trying to motivate the use of open sets for defining continuity (as in topology or metric spaces). I came to formulate the following definition of continuity for a function $f: X \rightarrow ...
0
votes
1answer
21 views

find the Classification of discontinuities of a function

In my assignment I have to find the Classification of discontinuities of the following function: $$f(x)=\frac{\sin^2(x)}{x|x(\pi-x)|}$$ I wanted to start with the value $x=0$ because the function ...
1
vote
2answers
30 views

The function is not continuous

$$C([a,b])=\{ f: [a,b] \to \mathbb{R} \text{ continuous} \}$$ $C([a,b])$ is a linear space. For $f \in C([a,b])$ we define $\|f\|_{\infty}:= \sup_{x \in [a,b]} |f(x)|$ and easily it can be shown ...
4
votes
4answers
64 views

Show that if $f$ is continuous at $a$ and $f(a)≠0$ then $f$ is nonzero in an open ball around $a$.

Here is the question I'm dealing with: Let $U$ be an open set of $\mathbb{R}^{n}$, $f:U\rightarrow\mathbb{R}^{n}$ a function and $a\in U$ a given point. Show that if $f$ is continuous at $a$ and ...
7
votes
2answers
113 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 ...
0
votes
4answers
34 views

Continuous function does not map closed set to closed set

I have a question in my textbook ask me to use this function $f(x)=x^2/(1+x^2)$ to show that continuous function does not necessarily map a closed set to a closed set. But I can't find any example to ...
3
votes
5answers
318 views

How is this example not a homeomorphism?

I am a beginner in Topology. I was going through Munkres book where I came across this example. The mapping $[0,1)\to S^1$ (unit circle) is bijective and continuous, but $f^{-1}$ is not continuous. ...
0
votes
2answers
28 views

If $\forall V\subseteq X$ where $x\in \overline V; f(x) \in \overline{f(V)}$, then $f$ is continous in $x$

Let $f:(X,\tau_X)\to (Y,\tau_Y)$ Prove: If $\forall V\subseteq X$ where $x\in \overline V; f(x) \in \overline{f(V)}$, then $f$ is continous in $x$. Could someone verify the following proof? ...
2
votes
2answers
47 views

Antiderivative is continuous

The following comes from Bass' book on Real Analysis: (Here $dy$ is Lebesgue measure) Exercise 7.6 Suppose $f:\mathbb{R}\to\mathbb{R}$ is integrable, $a\in \mathbb{R}$, and we define ...
1
vote
1answer
28 views

Why is the identity function from $\Bbb R$ with the Euclidean metric to $\Bbb R$ with the discrete metric not continuous?

Using only the definition of sequential continuity, show an example that $f(x) = x: \Bbb R \to \Bbb R'$ is not continuous, where $\Bbb R'$ has the discrete topology. So the definition of ...
1
vote
0answers
31 views

Lower semicontinuity on a metric space

I'm trying to prove something about lower semicontinuity for a map on a metric space $(X,d)$. I will try to write here my idea of the proof, hope someone can approve or contest it. Def. Let $(X,d)$ ...
3
votes
3answers
101 views

How to show that $\sum_{k=0}^{\infty} \frac{x^{k}}{k!}$ represents a continuous function

(This is a homework problem) I am trying to show that the series $\sum_{k=0}^{\infty} \frac{x^{k}}{k!}$ represents a continuous function on $\mathbb{R}$. My idea was to show that the functions ...
1
vote
0answers
20 views

Topological Embedding Which is Neither Open nor Closed

I'm having trouble coming up with an example of an embedding which is neither open nor closed. My attempts have included trying to find such a map from $\mathbb{R}$ (given the usual Euclidean ...
0
votes
1answer
29 views

Showing a function is not continuous at any other points.

I am having trouble picking a number x in the interval below. I need help picking the right interval. I tried sketching it, but I think I have trouble understanding it. Can someone help me clarify ...
0
votes
2answers
28 views

Injective implies invertible? Injective and well-defined implies bijective?

I have two questions regarding functions regarding linear maps: (Let $X$ and $Y$ be to Banach spaces) If $T:X\rightarrow Y$ is injective, then $T^{-1}$ exists, right? If $T:X\rightarrow Y$ is ...
2
votes
1answer
26 views

Topological field - Proving continuity of inversion

Given a field $F$ and an absolute value $|\ |$ on $F$, define the distance $d(x,y)$ between two elements $x,y\in F$ by $$ d(x,y) = |x - y|. $$ I just worked through the proofs that $d$ defines a ...
1
vote
5answers
63 views

Show that $f(x) = 0$ for all $x \in \mathbb{R}$ [duplicate]

$f: \mathbb{R} \to \mathbb{R}$ is continuous with $f(x)=0$ for all $x \in \mathbb{Q}$. Show that $f(x) = 0$ for all $x \in \mathbb{R}$. Can anyone please point me in the right direction as to how ...
0
votes
2answers
24 views

Is a function that has Holder order bigger than one constant?

I see from the Wikipedia that if a function $f$ over $[a,b]$ is Holder continuous with order strictly bigger than one, i.e. $$|f(x) - f(y)| < K |x-y|^\alpha$$ for some constant $K$ and ...
-1
votes
2answers
78 views

How $(f_n)$ converges uniformly on $[a, b]$

Let $(f_n)$ be defined and continuous on an interval $[a, b]$, and differentiable on $(a, b)$. Let $c \in [a, b]$. Assume that $(f_n(c))$ converges and that $(f'_n)$ converges uniformly on $(a, b)$. ...
1
vote
4answers
32 views

Epsilon-Delta Continuity proof (verification/help)

So, I am really bad at these problems, and I don't know why. Edit: The metric over $\Bbb R$ is assumed to be $|f(a,b)-f(x_1,x_2)|$ Problem statement: Define $f: \Bbb R^2 \rightarrow \Bbb R$ by ...