Investigate whether there exists a function $f : [a,b]\rightarrow R$ that is continuous and that takes exactly twice each of its values.
139763
A function takes every function value twice - proof it is not continuous
15487
Continuous function from a connected set?
102143
$f:\mathbb R\rightarrow \mathbb R$ be a continuous function such that $\int_{0}^{\infty}f(x)dx$ exists.
144614
If $f$ is continuous on $[a,b]$ and $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ .
74950
Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$
130019
Function that is continuous only on the transcendental numbers
54797
Find the continuous function defined over $[0,1]$ such that $\int_0^1f(x){\rm d}x=1, \int_0^1 xf(x){\rm d}x=a$ and $\int_0^1x^2f(x){\rm d}x=a^2 .$
36983
Does there exist a continuous function such that f(x) is rational for every irrational x, and f(x) is irrational for every rational x?
43189
Examples of continuous functions $f : \mathbb{R} \to \mathbb{R}$
62817
Functions which are Continuous, but not Bicontinuous
71487
Real Analysis: Continuity of a Composition Function
111485
Let $f(x)$ be continuous on [$0,1$] to $f(x)=0$ for each $x\in[0,1]$
97619
Which function ($f$) is continuous nowhere but $|f(x)|$ is continuous everywhere?
32571
Is the maximum function of a continuous function continuous?
2142
Finding a set of continuous functions with a certain property 2
26955
Continuous at exactly two points and differentiable at exactly one of them
75735
Let $f$ be a continuous function on an interval around $0$ and let $a_i=f(\frac{1}{i})$ (for large enough $i$)