If a function f has a derivative at x = c, which of the following statements is TRUE?

Question

I. Function f is continuous at x = C.

II. The limit of f at x = C is differentiable and exists.

III. Its graph cannot have a hole or jump at C.

IV. The tangent to the graph of f can be vertical at x = C.

Choices:

A. Statements I and II only C. Statements II, III, and IV only

B. Statements III and IV only D. Statements I, II, and III only

Answer 1

The answer is D, statements I, II, and III.

For a derivative to exist at a point x=c, we require that 1) the function have a value at c (f(x) is defined at x=c) and 2) the limit of f(x) must exist at f(c).

If there is a discontinuity, then either (1) or (2) fails: in a hole discontinuity, the function is not defined at that point and hence (1) fails, while in a step discontinuity, there is no limit and (2) fails (this is because the limit approaching from the left will differ from the limit approaching from the right). So, I must be true.

The limit must exist (because the definition involves the limit). Trivially, the limit is differentiable because f(c) is some constant at that point. So II is true.

III must be true because otherwise f(x) is not continuous at that point. Again, this is the most basic requirement for there to be a derivative.

Finally, because the derivative is the slope of the tangent line, an infinite slope does not exist and therefore IV cannot be true.