# 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) ...

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### If $f$ takes Cauchy sequence to Cauchy sequence then $f$ is continuous [duplicate]

If $f:X\to Y$ takes Cauchy sequence to Cauchy sequence then prove that $f$ is a continuous function. Let $x_n$ be a sequence in $X$ such that $x_n\to x\implies x_n$ is Cauchy $\implies f(x_n)$ is ...
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### If $f$ is a map from a topological space $Y$ to a metric space $X$, to prove that $f$ i

If $f$ is a map from a topological space $Y$ to a metric space $X$, to prove that $f$ is continuous at y, is it enough to show that for all $\epsilon >0$, there exists $V_{y}$ (neigbhourhood of $y$ ...
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### Check uniform equicontinuity of a function family

I am struggling to prove or disprove that the following function family is uniformly equicontinuous. $$F = \{f \in C^1[0,1]: \forall x \text{ } |f(x)| + \sqrt x |f'(x)| \leq 1 \}$$ First I tried to ...
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### Can we always multiply some function that is not differentiable everywhere with function that is to obtain differentiable product?

First of all, I think that before stating the general question it would be okay to make some concrete example of what do I have in mind. Let us take the function $f(x)=|x|$. We could write this ...
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### Continuous strictly increasing function is absolutely continuous iff set of infinite derivative maps to measure zero set

If $u:[a,b]\to\mathbb{R}$ is continuous and strictly increasing, prove that $u$ is absolutely continuous iff it maps $E:=\{x\in[a,b]:u'(x)=\infty\}$ into a set of measure 0. This question came from ...
### If $f:\mathbb R\to\mathbb R$ continuous does $f^{-1}$ also continuous?
Let $f:\mathbb R\to\mathbb R$ is bijective and continuous, does $f^{-1}$ is also continuous ? Does this result hold for $f:U\to\mathbb R$ where $U\subset \mathbb R^n$ ? and for $f:V\to\mathbb R^m$ ...