-1
votes
1answer
52 views

Calc I problem involving continuity [closed]

If $g(x)=(h(x))^2$, then give an example of a function $h:\Re\rightarrow\Re$ with the property that $h$ is not continuous at $x=0$ but $g$ is.
2
votes
2answers
41 views

Counterexamples for $f(\overline{A}) = \overline{f(A)}$ and $\overline{f^{-1}(B)} = f^{-1}(\overline{B})$ in (non-)continuous mapping $f: X \to Y$

Let $f$ be a mapping. Prove that the following three statements are equivalent. $f$ is continuous; $\forall A \subseteq X: f(\overline{A}) \subset \overline{f(A)}$; $\forall B \subseteq ...
1
vote
0answers
26 views

Basic calculus question with continuous function [duplicate]

This is actually not my question, it was asked yesterday by user176744 in this link $[0,n]$ continuous function problem and I feel as if it didn't get enough attention. I am also interested in a ...
4
votes
3answers
46 views

Does continuity of $f$ imply $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$?

I'm struggling to prove or disprove that the continuity of $f$ implies $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$. $f:X\to Y$ is a map between metric spaces $(X,d),(Y,d')$ while $\bar M$ denotes the ...
0
votes
1answer
55 views

Are there standard parameters for the Weierstrass nowhere differentiable function?

On Wikipedia, the Weierstass non-differentiable function is defined as: $$f(x)=\sum^{\infty}_{n=0}a^n\cos(b^n\pi x)$$ where $0<a<1$, $0<b$, and $ab>1+\frac 32 \pi$ Since it seems like, ...
2
votes
1answer
120 views

Equicontinuity of a pointwise convergent sequence of monotone functions with continuous limit

I was looking at this question, and trying to come up with a counterexample. After thinking about it, I thought the following might be true: Claim: let $\{f_n\}$ be a sequence of continuous, ...
1
vote
1answer
75 views

Proofs about continuity and convergence in topological spaces

I'm working on the following exercise: Let $f:(X,T)\to(Y,S)$ and $x\in X$. Prove that if $f$ is continuous at $x$ then if a sequence $\{x_n\}$ converge to $x$ we have $f(\{x_n\})\to f(x)$, show ...
0
votes
1answer
25 views

Function differentiable at one point and nowhere else continuous.

Is it possible to construct such a function? Just wondering. Specifically, I am thinking of $f:\mathbb{R}\to\mathbb{R}$ such that $f'(0)$ exists and $f$ is discontinuous for all ...
0
votes
2answers
33 views

Lipschitz does not imply fixed points

I have the following problem in mind: Let us say we have a function $f:X\rightarrow X$ (X is a complete metric space) and it respects that if $x\neq y$ then : $d(f(x),f(y))<d(x,y)$. My trouble ...
1
vote
1answer
82 views

$\exists f:\mathbb{R}\rightarrow \mathbb{R},$ continuous, non-constant, with uncountably many extrema?

I couldnt think of any; by intuition I don't think any can exist, but I can't figure out how to prove it. If it existed then the set of extrema would have to be uncountable but I think this might ...
4
votes
3answers
187 views

Prove or disprove that if $f$ is continuous function and $A$ is closed, then $\,f[A]$ is closed.

If $f:X\to Y$ is a continuous function and $A\subset X$ is a closed set, then $f[A]$ is also closed. I know that if $f[A]$ is closed implies $A$ is closed then $f$ is continuous, but I'm not sure ...
0
votes
0answers
58 views

Composition of uniform convergent function is not uniform convergent

I am trying to come up with an example for the following situation: Say we have 2 sequences of functions $f_n:U \rightarrow R $ and $g_n: \rightarrow W $ both uniformly convergent to $f$ resp. $g$ ...
7
votes
9answers
396 views

Nonpiecewise Function Defined at a Point but Not Continuous There

I make a big fuss that my calculus students provide a "continuity argument" to evaluate limits such as $\lim_{x \rightarrow 0} 2x + 1$, by which I mean they should tell me that $2x+1$ is a polynomial, ...
20
votes
9answers
735 views

Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true?

Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? It seems to me like they are equal definitions in a way. Can you give me a counter-example? Thanks
2
votes
2answers
98 views

Differentiability of a certain piecewise function

Consider the function $$ f(x)=\begin{cases} x & \textrm{if } x \textrm{ is rational} \\ -x & \textrm{if } x \textrm{ is irrational} \end{cases} $$ It is well-known that $f(x)$ is continuous at ...
7
votes
1answer
340 views

Dini's continuity vs Holder continuity

(listed items are just the definitions, you can skip to "Clearly" if you are familiar with them) Let $E \subset \mathbb{R}^N$ and let $f \colon E \to \mathbb{R}.$ The modulus of continuity of $f$ is ...
4
votes
4answers
142 views

Why Does the existence of $\frac{\partial f}{\partial x}$ not imply that $\frac{\partial f}{\partial x}$ is continuous?

For $f(x)$, the existence of $f'(x)$ implies the continuity of $f(x)$. And I am assuming that it also implies the continuity of $f'(x)$. My question is why in a function $g(x,y)$, is the existence ...
6
votes
2answers
156 views

Infinitely many zeros of a nonconstant continuous function?

Let $f:[0,1]\to\mathbb{R}$ be a nonconstant continuous function. Is $S=\{x: f(x)=0\}$ finite? I have thought of a function with countably many $0$'s like lots of triangular bumps at each point ...
17
votes
6answers
825 views

If $f$ is continuous at $a$, is it continuous in some open interval around $a$?

If $f: \mathbb{R} \to \mathbb{R}$ is continuous at $a$, is it continuous in some open interval around $a$?
7
votes
2answers
572 views

Continuity of the inverse $f^{-1}$ at $f(x)$ when $f$ is bijective and continuous at $x$.

Prove or disprove: Let $f:\mathbb{R}\to\mathbb{R}$ be bijective and $f$ is continuous at $x$. Then $f^{-1}$ is continuous at $f(x)$. Any hints are welcome. If this is false, I would like to have ...
3
votes
3answers
375 views

Does a monotone function on an arbitrary subset of $\mathbb R$ always have at most countable number of discontinuity?

I know a monotone function of a closed and bounded interval can have at most countably many point of discontinuity. And hence a monotone function on $\mathbb R$ can have at most countably many point ...
9
votes
2answers
1k views

Give an example of a function $h$ that is discontinuous at every point of $[0,1]$, but with $|h|$ continuous on $[0,1]$

Give an example of a function $h:[0,1]\to\mathbb{R}$ that is discontinuous at every point of $[0,1]$, but such that the function $| h |$ that is continuous on $[0,1]$. I don't really even know where ...
4
votes
1answer
321 views

Separately continuous functions that are discontinuous at every point

What are some good examples of separately continuous functions $f: X \times Y \rightarrow Z$ that are discontinuous at every point? Here's a theorem to rule out some spaces: link for a reference ...
2
votes
3answers
237 views

sequentially continuous on a non first-countable

Can you give me an example of a function which is sequentially continuous but not continuous? (I know that in first-countable spaces this is equivalent, but what about in spaces without this ...
3
votes
0answers
56 views

Differentiable function which is nowhere continuously differentiable [duplicate]

Possible Duplicate: How discontinuous can a derivative be? $x^2\cos(1/x)$ is the standard example for a differentiable function whose derivative is not continuous at $x=0$. But is there ...
3
votes
1answer
92 views

The subset of discontinuous in all points is not open in the space of bounded functions

Let $X\subset \mathcal{B}(\mathbb{R},\mathbb{R})$ be the subset of bounded functions $f:\mathbb{R}\to\mathbb{R}$ such that are discontinuous in all points. Prove that $X$ is not open (with usual ...