Mathematics

Problem

Let

f(x)=\begin{cases} ax^{2}, & x\le 1,\\ |x-1|+a, & x>1. \end{cases}

Which statement about the continuity and differentiability of

f

x=1

is true? a)

f

is continuous for all real

a

but never differentiable at

x=1

. b)

f

is continuous and differentiable at

x=1

for all real

a

. c)

f

is continuous for all real

a

and differentiable at

x=1

only when

a=\dfrac12

. d)

f

is continuous for all real

a

and differentiable at

x=1

only when

a=1