Let the function
$$f ( x ) = \begin{cases} x & \text { if } x \leq 0 \\ x \ln ( x ) & \text { if } x > 0 \end{cases}$$
a) ( 0.5 points) Study the continuity and differentiability of $f ( x )$ at $x = 0$. b) (1 point) Study the intervals of increase and decrease of $\mathrm { f } ( \mathrm { x } )$, as well as the relative maxima and minima. c) (1 point) Calculate $\int _ { 1 } ^ { 2 } f ( x ) d x$.