# Tagged Questions

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### Can someone explain the concept of continuity and differentiability for functions of several variables?

Can someone explain the concept of continuity and differentiability for functions of several variables? Illustrated examples will definitely help, on how to solve problems(or establish proofs) of the ...
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### A bijective function $f$ between two compact Hausdorff spaces is continuous if $f$ preserves compact sets [duplicate]

I am trying to prove that if $f: X \longrightarrow Y$ is a bijection between two compact Hausdorff spaces such that $f[W]$ is compact in $Y$ for all compact $W$ in $X$, then $f$ is continuous. Here ...
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### A continuous surjective function from $(0,1]$ onto $[0,1]$

I'm trying to construct a continuous surjection from $(0,1]$ onto $[0,1]$, but I'm not getting anywhere. I don't immediately see a contradiction which falsifies the existence of such a function, so my ...
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### Is the standard part function another devil's staircase?

The devil's staircase or Cantor function is an awesome function that increases value but has derivative zero everywhere (or "almost", whatever that means). I was incredibly amazed when I found out ...
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### Conditions yielding a unique fixed point of a continuous differentiable function.

Let $f$ be a function defined on $[0,1]$ which is continuous for each point in $[0,1]$ and differentiable for each point in $(0,1)$. Suppose that $f^\prime (x) \neq 1$ for every $x \in (0,1)$. ...
When using and examining Kakutani's fixed-point theorem, I've got a question about upper hemicontinuity. A correspondence $f:X\rightarrow2^Y$ is a point-to-set mapping. One way to define upper ...