Continuous function, that satisfies $f(f(x)) = x$ on $[0, 1]$
113687
$f$ is continuous function from $ [0,1]$ to itself and f(0)=0, f(1)=1 and $f(f(x))=x$ then show that $f(x)=x$ on$[0,1]$
17234
Show that a continuous function has a fixed point
11158
Proof that a continuous function with $f(x) = f(x^2)$ is constant.
38257
Continuous bijection from $(0,1)$ to $[0,1]$
14251
$P[X=x]=0$ when $X$ is continuous variable
68165
Find a function $f: \mathbb{R} \to \mathbb{R}$ that is continuous at precisely one point?
409
functions satisfying $f(x - y) = f(x) f(y) - f(a - x) f(a + y)$ and $f(0)=1$
14578
Proving that an additive function $f$ is continuous if it is continuous at a single point
148266
Let $f:X\to X $ be continuous. Does $f $ have a fixed point when $X=[0,1)$ or $X=(0,1) $?
29582
Continuous mapping $f: [0,1]\rightarrow (0,1)$ CSIR December $2013$
130019
Function that is continuous only on the transcendental numbers
103241
A function that is continuous at the origin but not differentiable there
77943
If $f$ is continuous, then there exists $x+\frac{1}{n}\in[0,1]$ and $f\left(x+\frac{1}{n}\right)=f(x)$
17343
Function on $[a,b]$ that satisfies a Hölder condition of order $\alpha > 1 $ is constant
58816
Existence of a continuous function which does not achieve a maximum.
5419
Continuous functions with $f^2(x)=g^2(x)\neq 0$
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. $f(x) = f(x+\frac{1}{2})$