Mathematics

Problem

Consider the function

f

defined by

f(x)=\begin{cases} x^2, & x<1,\\ 2x-1, & x\ge1. \end{cases}

Respond to the prompts below regarding differentiability at

x=1

. (a) Verify that

f

is continuous at

x=1

. (b) Determine

\displaystyle\lim_{x\to1^-}\frac{f(x)-f(1)}{x-1}

. (c) Determine

\displaystyle\lim_{x\to1^+}\frac{f(x)-f(1)}{x-1}

. (d) Decide whether

f

is differentiable at

x=1

. Support your conclusion with the computations from parts (b) and (c).