0
votes
2answers
23 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 ...
0
votes
1answer
74 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 ...
0
votes
0answers
49 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
341 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, ...
17
votes
9answers
454 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
72 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
239 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 ...
3
votes
4answers
130 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
149 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
785 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
434 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
2answers
241 views

Does a monotone function on an arbitray 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 ...
8
votes
2answers
729 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
225 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
132 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
47 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
90 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 ...