Let $f(x)$ be defined on $R$ by $f(x)=2x-1$ if if and $f(x)=2x$ if $x1.$ Define $g(x)={\displaystyle \underset{0}{\overset{x}{}}}f(y)dy.$

a$)$ Show that $g(x)$ is differentiable at all $x$ other than $x=1,$ and that $g^{\text{'}}(1)$ does not exist.

b$)$ Explain why the result of part a$)$ does not contradict the Fundamental Theorem of Calculus.