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#

1 Answer
Nov 4, 2016

(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#.