If the derivative of a continuous function f is undefined at x = c, f can be increasing at x = c.

Question

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Answer 1

It possible. For example f(x) = IxI at x = 0 not differentiable but it increasing at x = 0 for x ≥ 0

Answer 2

Of course it can still be increasing. Think about the following example:

f(x) = x, if x < 0 and 2x, x ≥ 0

The function is continuous and the derivative at x = 0 is not defined. Nevertheless, the function is monotonically increasing throughout (-∞,∞).

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