Differentiability of a Piecewise Function

Let the function $g$ be defined by $\displaystyle g(x) = \begin{cases} \displaystyle x^2 - 2x + 1 & \text{if } x < 1 \\ \displaystyle e^{x-1} - 1 & \text{if } x \geq 1 \end{cases}$. Which of the following statements about $g$ are true?

I. $\displaystyle \lim_{x \to 1^-} g(x) = \lim_{x \to 1^+} g(x)$

II. $\displaystyle \lim_{x \to 1^-} g'(x) = \lim_{x \to 1^+} g'(x)$

III. $g$ is differentiable at $x = 1$.