Prove that $f(\frac{x+y}{2}) \le \frac{f(x)+f(y)}{2}$ is continuous almost everywhere [closed]

Let $f:(a,b) \rightarrow \mathbb{R}$ be a bounded measurable function, and $f(\frac{x+y}{2}) \le \frac{f(x)+f(y)}{2}$. Take a sequence of function $\{j_n\}$,$j_n \in C^\infty$, $j_n \ge 0$, supp $j_n \subset (-\frac{1}{n},\frac{1}{n}) $ and $\int j_n = 1$. Prove that $\forall \delta > 0$, $\{j_n * f\}$ is uniformly continuous on $(a+\delta, b-\delta)$. And prove that $f$ equals to a continuous function $g$ almost everywhere.

$j_n * f$ is convolution.

Both questions are difficult for me.