Answer
a) Any value of $k$ allows $f$ to be continuous.
b) $k=2$
Work Step by Step
a) $f$ is continuous when the limit from the left and right equal the value of $f$ at $1$.
$f(1) = 1^1-1 = 0$
$\lim_{x \to 1^-} f(x) = 0$
$\lim_{x \to 1^+} f(x) = k(0)$
Thus, any value of $k$ will make the limit from the right equal the limit from left and the point at $x=1$.
b) $f$ is differentiable when the limit of the derivative from the left and right equal each other:
$\lim_{x \to 1^-} f'(x) = 2x = 2(1) = 2$
$\lim_{x \to 1^+} f'(x) = kx = = k(1) = k$
Thus, $k=2$ allows for $f$ to be differentiable.
