Question

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->0)(f(2+h)-f(2))/h=5`

Answer 1

(C)

Explanation:

Noting that a function `f` is differentiable at a point `x_0` if

`lim_(h->0)(f(x_0+h)-f(x_0))/h = L`

the given information effectively is that `f` is differentiable at `2` and that `f'(2) = 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(x) = {(x^2sin(1/x) if x!=0),(0 if x=0):}`, which is differentiable at `0`, but whose derivative has a discontinuity at `0`.