Proving that $f(x+y) = f(x) + f(y)$ and $f$ being continuous at one point implies that $f$ is continuous on ${\bf R}$
14578
Proving that an additive function $f$ is continuous if it is continuous at a single point
104369
If $f$ continuous and $f(x^2) = f(x)$, then $f$ is a const
69122
If $f,g$ are uniformly continuous prove $f+g$ is uniformly continuous but $fg$ and $\dfrac{f}{g}$ are not
74502
Proving the bijection of a continuous function $f$ such that $|f(x)-f(y)| \geq \log(1+|x-y|) \forall x,y$ in $\mathbb{R}.$
78834
if f(x) if differentiable and continuous, prove $\frac{af(a)-bf(b)}{a-b} = f(c) + cf'(c) $
152841
I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive.
28289
Show that $Kf(x,y)=\int_0^1k(x,y) f(y) \,dy\\$ is linear and continuous
94781
Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$
54729
Prove that $f$ has a fixed point .
132013
How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
37966
If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$
43874
Differentiability implies continuity - A question about the proof
19916
$f(x)=f(x^2+ 1/4)$ , $f$ is continuous from $\mathbb{R}$ to $\mathbb{R}$
46629
Prove if $f(x) = g(x)$ for each rational number x and $f$ and $g$ are continuous, then $f = g$
68048
Prove that if $f$ is differentiable on $[a,b]$ and $f'$ is monotonic on $[a,b]$ , $f'$ is continuous on $[a,b]$
90565
If $f:[0,1] \rightarrow \mathbb{R}$ is continuous function such that $f(0)=f(1)$ then there exists $ x\in [0,1]$ such that $f(x) = f(x+\frac{1}{n})$.
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$