Let us now deduce the condition(s) on the derivative of a function which determines whether is increasing/decreasing on a given interval. We are assuming that is everywhere differentiable.
The function on the left side, , is a strictly increasing function. Notice that the slope of the tangent drawn at any point on this curve is always positive. Hence, a sufficient condition for to be strictly increasing on a given domain is
Later on, we will see that this is not a necessary condition for a function to be strictly increasing.
In Fig-, the function on the right side, is not strictly increasing though it is increasing. Notice that or is never negative. Hence, a sufficient condition for to be increasing on a given domain is
Note that for these condition on the derivatives to be applied, the function must be differentiable in the given domain. However, these conditions will hold good even if the function is non differentiable, but only at a finite number (or infinitely countable number) of points. For eg, is strictly increasing on . However, is non-differentiable at all integers (a countable set).
A function must be continuous for these conditions to be applied. Consider . This is non-differentiable (due to discontinuities) at all integers. At all other points, . However, we know that is not strictly increasing. Similarly, is non-differentiable (and non-continuous) at . At all other points, so that should be strictly decreasing on . However, it is not strictly decreasing on although it is strictly decreasing in the separate intervals and . Therefore, we see that discontinuous functions cannot be subjected to the derivative condition even though they may be discontinuous only at a finite (or infinitely countable) number of points.
Now consider and in Fig
Extending the previous case, we get the conditions for a (strictly) decreasing function:
The remarks made for the increasing case hold true here also.
Before concluding this section, here are some other facts worth paying attention to:
(a) If is strictly increasing, then is also strictly increasing. Similarly, if is strictly decreasing then is also strictly decreasing.
(b) If and have the same monotonicity (both increasing or decreasing) on , then and are monotonically increasing on .
(c) If and have opposite monotonicity on , then and are strictly decreasing on
(d) The inverse of a continuous function is continuous
(e) If except for a finite (or an infinitely countable) number of points where is still strictly increasing on . This is why we said earlier that is not a necessary condition for strict increase. For example, in a later example we will consider the graph of the function . We will see that is not always positive ; even then, increases strictly, because the points at which are countable
(f) Similarly if except for a finite (or an infinitely countable) number of points where is still strictly decreasing on