Property of a continuous function with $f(0)=f(2)$
12632
Universal Chord Theorem
16735
If $f,g:X \to Y$ are continuous and $Y$ is $T_2$, then $\{x \in X\,|\,f(x)=g(x)\}$ is closed
2142
Finding a set of continuous functions with a certain property 2
104369
If $f$ continuous and $f(x^2) = f(x)$, then $f$ is a const
44574
Continuity of absolute value of a function
154881
Question on differentiability of a continuous function
83338
Show that there's no continuous function that takes each of its values $f(x)$ exactly twice.
58664
Proving a function is continuous for a fixed variable
128911
Continuous extension of a real function
10747
Given function $g$ continuous in line $[0,\infty)$...
95970
Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. If $f(0)=f(1)$ then exists $x\in [0,1]$ s.t. $f(x) = f(x+\frac{1}{2})$
151410
Inequality of continuous functions
19916
$f(x)=f(x^2+ 1/4)$ , $f$ is continuous from $\mathbb{R}$ to $\mathbb{R}$
150554
proof that $f'$ is continuous at $a$
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 $\int_0^1x^2f(x){\rm d}x=a^2 .$
130023
Prove that the sum of two continuous functions is continuous
145919
If $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous at $0$ and $f(x)=f(2x)$ for each $x\in\Bbb{R}$ then $f$ is constant.