Suppose And $\lim _ { x \rightarrow 1 ^ { + } } \frac { f ( x ) - f ( 1 ) } { x - 1 } = 1$

Multiple Choice

Suppose $f ( 1 ) = 2 , \lim _ { x \rightarrow 1 ^ { - } } f ( x ) = 2 , \text { and } \lim _ { x \rightarrow 1 ^ { + } } f ( x ) = 2 , \lim _ { x \rightarrow 1 ^ { - } } \frac { f ( x ) - f ( 1 ) } { x - 1 } = - 2$ and $\lim _ { x \rightarrow 1 ^ { + } } \frac { f ( x ) - f ( 1 ) } { x - 1 } = 1$ Is $f$ continuous and/or differentiable at $x = 1 ?$

A) f is not continuous but differentiable at x = 1
B) f is neither continuous nor differentiable at x = 1
C) f is continuous at but not differentiable at x = 1
D) f is both continuous and differentiable at x = 1