Does $f(\frac{x+y}{2})\leqslant \frac{f(x)+f(y)}{2}$ imply continuity?
20614
Example of a function such that $\varphi\left(\frac{x+y}{2}\right)\leq \frac{\varphi(x)+\varphi(y)}{2}$ but $\varphi$ is not convex
129856
Show that $f(x)=\frac{x}{1+|x|}$ is uniformly continuous.
22303
How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero?
60521
Continuity of a convex function
131485
Continuity of $\frac{x^3y^2}{x^4+y^4}$ at $(0,0)$?
98921
Does $y(y+1) \leq (x+1)^2$ imply $y(y-1) \leq x^2$?
102457
for any $A\subseteq X$, $f(\overline{A})\subseteq\overline{f(A)}$ , if and only if $f: X \to Y$ is continuous.
136011
Must $f$ be continuous?
32571
Is the maximum function of a continuous function continuous?
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
56567
Prove or disprove the uniform continuity of $f: \mathbb{C} \to \mathbb{C}, z \mapsto f(z)=\frac{z^2}{1+|z|}$
127114
Differentiability implying continuity
58664
Proving a function is continuous for a fixed variable
151573
Show f is continuous on (0, ∞) iff f is continuous at 1 for a defined function.
37850
Does the everywhere differentiability of $f$ imply it is absolutely continuous on a compact interval?
143618
If a function $f$ is continuous $\implies$ $|f|$ is continuous.(Answered by Myself)
141223
$f(x^2) = 2f(x)$ and $f(x)$ continuous