# Questions tagged [continuity]

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) or (general-topology)

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### Let f : R → R be continuous, such that lim f(x), x→−∞ and lim f(x), x→+∞ exist and are real numbers. Prove that f is uniformly continuous

For f as x tends to -∞ , I have this: Consider ε > 0. Because f converges to a limit in −∞, we know that given ε> 0, there exists an M such that for all x < M, |f(x)-L| <= ε/2, (where lim ...
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### If $f$ is differentiable in a neighbourhood of $c$, is $f'(x)$ continuous at $x=c$?

I'm pretty sure the statement "If $f$ is differentiable at $c$, is $f'(x)$ continuous at $x=c$" is wrong and there are quite a lot of counterexamples to that, but what if we add a " ...
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### Domain and Continuity of piecewise composite function

Consider the function $$g(x)=\begin{cases} e^{-x}& \text{ if } x\geq 0 \\ \sqrt{\left | x \right |}& \text{ if } x< 0 \end{cases}$$ and $$h(x)=(x-4)(x+1)^2$$, when is the composite ...
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### Continuous and one-one function that is bounded

Actually option 2 is given as answer.So,other options are incorrect.I got an counter example for other options that they are not true.But I am unable to find a counter example to show option 4 is ...
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### Continuity of the derivative function. [closed]

Suppose we have a function f(x) that is differentiable for all values of x. Is it necessary for the derivative function to be continuous for all x ?.
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### Show that if $f_n$ converges uniformly on $[0,1]\cap\mathbb{Q}$, then $f_n$ converges uniformly on $[0,1]$.

The question is as follows Let $f_n$ be a sequence of continuous functions defined on $[0,1]$. Suppose that $f_n$ converges uniformly on $[0,1]\cap \mathbb{Q}$ to $f$. Show that $f_n$ converges ...
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### Elimination of a candidate of optimal solution based on SOC

A function $f(x; \alpha)$ is continuous in both variable $x$ and parameter $\alpha$,first and second partial derivatives of $f$ are continuous as well. I want to find the optimal solution of the ...
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### Continuity of a function at points where it became indeterminate form

If a function $f(x)$ at point let's say at $x=a$, $f(a)$ is an indeterminate($\frac{0}{0}$ etc. ) form then 1.is it sufficient to say that function will be discontinuous at $x=a$ Even though limit ...
Hi guys I'm new to topology and was asked to prove the following, of which I am having troubles with: Let $F:X \times I \rightarrow Y$ be a continous function. For each $t \in I$ let $f_{t} = F(x,t)$. ...
Let $f:X\to Y$ be a continuous function (where $X$ and $Y$ are topological spaces). Let $\lim_{n\to\infty}x_n=x$. I wanted to prove that $f(x_n)$ converges to $f(x)$ as $n$ goes to infinity. Here is ...