Let $f : [ a , b ] \rightarrow \mathbb { R }$ be a piecewise continuous function taking values in an interval $J$. Let $\varphi$ be a continuous and convex function on $J$. Prove that
$$\varphi \left( \frac { 1 } { b - a } \int _ { a } ^ { b } f ( t ) \mathrm { d } t \right) \leqslant \frac { 1 } { b - a } \int _ { a } ^ { b } \varphi \circ f ( t ) \mathrm { d } t .$$
You may use Riemann sums.