Assume that $f : A \to A$ is a function such that $\forall x, y \in A,\|f(y)-f(x)\|=\|y-x\|$. Prove that $f(A)= A$
62479
A isometric map in metric space is surjective?
105319
A calculus problem with functions such that $f''(x) = g(x)$ and $g''(x) = f(x)$
85739
Show that $f(x) = 0$ for all $x \in [a,b]$ given $|f'(x)| \leq C|f(x)| $
48592
Prove that $f'$ exists for all $x$ in $R$ if $f(x+y)=f(x)f(y)$ and $f'(0)$ exists
64905
Let $f \in C [a,b]$ such that $\forall x\exists y$ such that $|f(y)| \leq \frac{1}{2}|f(x)|$. Prove that $\exists c\in [a,b]$ such that $f(c) = 0$.
28007
Assuming: $\forall x \in [0,1]:f(x) > x$ Prove: $\forall x \in [0,1]:f(x) > x + \varepsilon $
83894
$f(x^2) = xf(x)$ implies that $ f(x) = mx$?
152841
I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive.
47135
Prove if a = b, then f(a) = f(b) for any function f (with natural deduction)
60348
How do I prove $F(a)=F(a^2)?$
104044
Prove continuous functions $f$ and $g$ are equal
65946
A function $f: \mathbb R \to \mathbb R$ satisfies equation $f(x+y)=f(x)f(y)$ for all $x,y\in \mathbb R $
97184
Show: $ f(a) = a,\ f(b) = b \implies \int_a^b \left[ f(x) + f^{-1}(x) \right] \, \mathrm{d}x = b^2 - a^2 $
34343
Determine all $f : \mathbb R \rightarrow \mathbb R$ such that $f(f(x) + y) = f(x^2 - y) + 4yf(x)$.
1071
Determine all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $xf(y)+yf(x)=(x+y)f(x^2+y^2)$ for all $x,y\in\mathbb{N}$ (contest question)
74502
Proving the bijection of a continuous function $f$ such that $|f(x)-f(y)| \geq \log(1+|x-y|) \forall x,y$ in $\mathbb{R}.$
13253
Prove $f(S \cup T) = f(S) \cup f(T)$