# 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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### Does Riemann integral of everywhere continuous and nowhere differentiable functions (with chosen values at the boundary points) can attain any value?

Suppose that we choose some interval and fix it, for example let us choose interval $[0,1]$. If $f$ is some everywhere continuous and nowhere differentiable function defined on $[0,1]$, then, because ...
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### Is the Restriction of a Continuous Map again a Continuous Function?

Is true that if $g:Y\to Y$ is continuous then a mapping $f:X\to Y$ with $X \subset Y$ is continuous? I think it's true. Since for every open set $U$ in $Y$, we have that $g^{-1}(U)$ is open in $Y$. ...
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### If $S$ is not compact, there is a continuous function unbounded on $S$

problem This was given to me as a homework problem to prove: If $S \subseteq \mathbb{R}$ is not compact, then there exists a continuous function $f : S \rightarrow \mathbb{R}$ that is unbounded ...
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### Constructing a sequence of functions, not Cauchy

I'm working in the set $B = \{ f \in C[0,1] : \int_0^1 f(x)dx \leq 1\}$. I'm constructing an argument to show that there exists at least one sequence that has a subsequences satisfying the property ...
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### “Continuous maps are those maps that do not tear space apart”

In a tutorial I wanted to give a quick explanation of the property of continuity. One of the common intuitions for continuity is that it preserves connection: Continuous maps do not map connected ...
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### Linear programming is continuous

Consider an arbitrary linear program: $$\max \vec c \cdot \vec x$$ subject to: $$\textbf{A}\cdot \vec x = 0, \quad \vec a \le \vec x \le \vec b$$ Assume that this program is feasible and bounded. ...
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### Using $\epsilon-\delta$ proof to prove continuity

Use an $\epsilon-\delta$ proof to show that $f : R \setminus \left \{ \frac{-3}{2} \right \} \rightarrow R$ , $$f(x) = \frac{3x^2-2x-5}{2x+3}$$ is continuous at $x = -1$ Hello there. Can anyone ...
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### Rigor in proving continuity of $f$ over a closed interval $I$

Given a function $f$ on a closed interval $I \subset \mathbb{R}$, where $I = [a,b]$, to prove continuity of $f$ over the interval $I$, what is generally done is the following. 1. We prove that $f$ is ...
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### A smooth nowhere analytic function such that all derivatives are monotone

Related questions that might provide some context: (1) (2) (3) (4) Let's restrict our attention to real-values functions on an open unit interval $f:(0,1)\to\mathbb R$. There are examples $\!^{[1]}$...
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### $M,N$ metric spaces, $\phi:M\to N$ a surjective open map. Show that the map $f:N\to P$ is continuous iff $f\circ \phi$ is continuous
I need to show the following: $M,N$ metric spaces, $\phi:M\to N$ a surjective open map. Show that the map $f:N\to P$ is continuous iff $f\circ \phi$ is continuous In order to show that the composite ...
### Continuity and differentiability of $f(x)=\sqrt{1-x}+\sqrt{x-1}$ [closed]
Given a function $f(x)=\sqrt{1-x}+\sqrt{x-1}$. Is $f(x)$ a continuous function. Is $f(x)$ a differentiable function.