Prove that there exist discontinuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y) = f(x) + f(y)$.
5373
Additive function $T: \mathbb{R} \rightarrow \mathbb{R}$ that is not linear.
79107
Give an example of a function $f: \mathbb{R} \to \mathbb{R}$ which is continuous only at $0$.
105455
Prove that "injective function $f:X\to Y$ exists" and "surjective function $g:Y\to X$ exists" is logically equivalent.
37976
Find all functions $f$ such that $(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$
125788
A function with a property $f(x+y)=f(x)f(y)$
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)
140445
Sequence of continuous functions whose pointwise limit is discontinuous
66391
Show that $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable
24810
Functions satisfying $(b-a)f'(\tfrac{a+b}{2}) = f(b)- f(a)$
15820
Discontinuous linear functional
11453
Find a real function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$?
105652
Suppose $f: \mathbb{R} \to \mathbb{R}$ is function such that $f^3$ is continuous on $\mathbb{R}$. Prove that $f$ is continuous on $\mathbb{R}$
67866
If $f:\mathbb{R}\to\mathbb{R}$ is a left continuous function can the set of discontinuous points of $f$ have positive Lebesgue measure?
67848
Let $A$ be the set of all functions $f:\mathbb{R}\to\mathbb{R}$ that satisfy the following two properties.
34077
Let $f:\mathbb{R}\to\mathbb{R}$ satisfy $f\big(x-f(y)\big)=f\big(f(y)\big)+x\cdot f(y)+f(x)-1$ for all $x,y\in\mathbb{R}$.
149354
Proof of continuity $f(x+y) = f(x) + f(y)$
155923
continuity of $f(x,y)$ in $\mathbb R^2$.