# Let f be a function so that (below). Which must be true? I. f is continuous at x=2 II. f is differentiable at x=2 III. The derivative of f is continuous at x=2 (A) I (B) II (C) I & II (D) I & III (E) II & III

## ${\lim}_{h \to 0} \frac{f \left(2 + h\right) - f \left(2\right)}{h} = 5$

Nov 4, 2016

(C)

#### Explanation:

Noting that a function $f$ is differentiable at a point ${x}_{0}$ if

${\lim}_{h \to 0} \frac{f \left({x}_{0} + h\right) - f \left({x}_{0}\right)}{h} = L$

the given information effectively is that $f$ is differentiable at $2$ and that $f ' \left(2\right) = 5$.

Now, looking at the statements:

I: True

Differentiability of a function at a point implies its continuity at that point.

II: True

The given information matches the definition of differentiability at $x = 2$.

III: False

The derivative of a function is not necessarily continuous, a classic example being $g \left(x\right) = \left\{\begin{matrix}{x}^{2} \sin \left(\frac{1}{x}\right) \mathmr{if} x \ne 0 \\ 0 \mathmr{if} x = 0\end{matrix}\right.$, which is differentiable at $0$, but whose derivative has a discontinuity at $0$.