Show that a function that satisfies $f(\frac{x+y}{2}) \leq \frac{f(x)+f(y)}{2}$ is continuous
53627
Proving continuity of $f$
71049
Functional equation $f(xy)=f(x)+f(y)$ and continuity
144001
If $f$ is continuous and $\,f\big(\frac{1}2(x+y)\big) \le \frac{1}{2}\big(\,f(x)+f(y)\big)$, then $f$ is convex
137043
To show that function is constant
145368
How to show $f:Y\to X^{+}$ is continuous?
98772
Continuity of a function in $\mathbb{R}$
11158
Proof that a continuous function with $f(x) = f(x^2)$ is constant.
110112
Proving a function is Lipschitz continuous
12408
Prove that a function whose derivative is bounded is uniformly continuous.
58629
Show that the $\max{ \{ x,y \} }= \frac{x+y+|x-y|}{2}$.
58664
Proving a function is continuous for a fixed variable
127449
Uniform continuous of $f(x,y) = \sin(x^2+y)$
57180
Techniques to prove a function is uniformly continuous
154572
Continuity of a function at an isolated point
125682
Proof verification: the product of two continuous functions is continuous
14495
I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ which are continuous and satisfy $f(x+y)=f(x)+f(y)$
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 $\mathbb R$