If $f$ is a continuous function and $f(a + b) = f(a) + f(b)$, how do I prove that $f(x) = mx$ where $m=f(1)$?
13086
Overview of basic facts about Cauchy functional equation
77267
Given $f$ is continuous and $f(x)=f(e^{t}x)$ for all $x\in\mathbb{R}$ and $t\ge0$, show that $f$ is constant function
40209
$f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
122160
$f:\mathbb R \to \mathbb R$ be a continuous such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , is $f(x)=x , \forall x \in \mathbb R$?
75266
$f(a+b)=f(a)+f(b)$ but $f$ is not linear
50552
find functions that $f(m+k)=f(mk-n)$
154034
Suppose that $f(x)$ is continuous on $(0, \infty)$ such that for all $x > 0$,$f(x^2) = f(x)$. Prove that $f$ is a constant function.
131992
A problem on continuous functions
105700
Proving $\,f$ is constant.
113975
Let $f:\mathbb{R}\to\mathbb{R}$, $f(kx)=f(x)$. Show that if $f$ is continuous at $0$ then $f$ is a Constant function
141223
$f(x^2) = 2f(x)$ and $f(x)$ continuous
118207
Prove that the product of two continuous functions is continuous
130023
Prove that the sum of two continuous functions is continuous
117502
Proving that a certain continuous function is surjective.
14495
I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ which are continuous and satisfy $f(x+y)=f(x)+f(y)$
81720
$f(x) + f(x+f(y)) = y + f(f(x) + f(f(y)))$
158888
Prove that the function $f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$.