# Tagged Questions

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) ...

30 views

53 views

### Understanding the notation $f\colon D\subseteq\mathbb{R}\to\mathbb{R}$ in the context of uniform continuity

In brief, I've had virtually no mathematical education prior to seven months ago. What I did know had largely been the result of watching Khan Academy. Seven months ago I began a precalculus textbook, ...
42 views

67 views

### Given $A$ a symmetric square matrix. Why is $f_A: O(n) \to \mathbb{R}^{n \times n}: Q \mapsto Q^{\top} A Q$ continuous?

Given $A$ a symmetric square matrix. And $O(n)$ the set of orthogonal matrices. Why is the following function continuous: $$f_A: O(n) \to \mathbb{R}^{n \times n}: Q \mapsto Q^{\top} A Q$$ This is ...
19 views

32 views

### Confusion about Differentiability with partial derivatives

Theorem: Let $A ⊂ R^n$ be open and $f: A ⊂ R^n → R^m$. Suppose $f = (f_1,...,f_m)$. If each of the partials $\frac{∂f_j}{∂x_i}$ exists and is continuous on $A$, then $f$ is differentiable on A. ...
45 views

### A Function is indefinitely differentiable

Suppose $f(x)=e^{-\frac{1}{x-a}-\frac{1}{x-b}}$ if $x \in (a,b)$, and $f(x)=0$ otherwise, show that $f \in C_{c}^{\infty} (R)$ I think I have to just find the general form of the $n^{th}$ derivative ...
33 views

35 views

### Does local Lipschitz continuity on a compact metric space imply global Lipschitz continuity? [duplicate]

I've got two problems on this issue: If $X,Y$ are metric spaces, $X$ compact, suppose $f:X\to Y$ is locally Lipschitz continuous on $X$, then is it also globally Lipschitz continuous? My guess ...
87 views

### Show that the derivative of a function is not continuous

$$g(x)=\begin{cases} x+2x^2\sin\left(\frac{1}{x}\right)&\text{ if }x\neq0\\\ 0&\text{ if }x=0 \end{cases}$$ Show that there is a sequence $\{x_n\}$ with $\{x_n\} \to 0$ as $n$ approaches ...
64 views

105 views

### Proving that a one-to-one continuous function on a compact subset has a continuous inverse

This is a curious problem I found in the "challenge" section of the text I'm learning real analysis from. Suppose $A$ is a compact subset of $\mathbb{R}^{n}$ and that $f$ is a continuous function ...