If $f$ is continuous $\exists t: f(t)=\frac{f(x_1) + ... + f(x_n)}{n}$
69219
Prove there is a point $z\in[a,b]$ at which $f(z)=\frac{f(x_1)+f(x_2)+\cdots+f(x_k)}{k}$
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 $f(x) = f(x+\frac{1}{n})$.
110458
Show that if $f$ is continuous on $(0,1)$ and $\lim_{x \to 0}f(x) = + \infty$ then $f$ is not uniformly continuous
113740
Find a uniformly continuous function such that $a_{n+1}=f(a_n)$
106060
If $f$ is continuous and with a limit at infinity then $f$ is uniformly continuous
129579
If $f''(x_0)$ exists then $\lim_{x \to x_0} \frac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0)$
14067
If f uniformly continous then $|f(x)|\leq C\cdot (1+|x|)$ for constant C
50012
$f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$
14578
Proving that an additive function $f$ is continuous if it is continuous at a single point
131992
A problem on continuous functions
74950
Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$
7876
If $f(x+y)=f(x)+f(y) ,\forall\;x,y\in\Bbb{R}$, then if $f$ is continuous at $0$, then it is continuous on $\Bbb{R}.$
111290
If $f$ continuous and $\lim_{x\to-\infty }f(x)=\lim_{x\to\infty }f(x)=+\infty $ then $f$ takes its minimum.
53627
Proving continuity of $f$
13927
Values of $a$ s.t. for all continuous $f$ with $f(0)=f(1)$ there exists $x$ s.t. $f(x+a) = f(x)$
50111
non-continuous function satisfies $f(x+y)=f(x)+f(y)$
86327
If the absolute value of a function is continuous, is the function continuous?