If a function's derivative is positive on an interval, it is strictly increasing throughout that interval. Using the mean value theorem, two equal or decreasing values would force f′(c) ≤ 0, contradicting the assumption. Similarly, a negative derivative throughout an interval implies a strictly decreasing function.
If the derivative is positive throughout an interval, except for a finite number of points where it is 0, the function is still strictly increasing. It is increasing on every interval between the zero derivatives, and it increases through each zero derivative. If it decreased across a zero derivative, we would again invoke the mean value theorem to produce a contradiction.