Exercise 7.5.1

(a)

Let $f(x)=|x|$ and define $F(x)={\int }_{-1}^{x}f$. Find a piecewise algebraic formula for $F(x)$ for all $x$. Where is $F$ continuous? Where is $F$ differentiable? Where does $F^{\prime }(x)=f(x)$ ?

(b)

Repeat part (a) for the function $$f(x)=\{\begin{array}{cc}1\hfill & \text{ if }x<0\hfill \\ 2\hfill & \text{ if }x\ge 0\hfill \end{array}$$