Mathematics

Problem

A function

f

is defined by

f(x) = \begin{cases} kx^{2}+3, & x\le 1, \\ 2x+k, & x>1. \end{cases}

The constant

k

is a real number. (a) Determine the value(s) of

k

for which

f

is continuous on

(-\infty,\infty)

. Show all work leading to your result. (b) For the value(s) of

k

obtained in part (a), justify that

f

is continuous for every real number by referring to appropriate properties of the component functions.