Limits Continuity And Differentiability

Let \(f:(a,b) \to [1,\infty )\) be a continuous function and g : R \(\to\) R be defined as \(g(x) = \left\{ {\matrix{ 0 & , & {x < a} \cr {\int_a^x {f(t)dt} } & , & {a \le x \le b} \cr {\int_a^b {f(t)dt} } & , & {x > b} \cr } } \right.\) Then,