5.4 Monotonic functions

The term “monotonic” conveys the meaning of maintaining order or the sense of “no change”. In the context of function, we think a monotonic function as the one whose successive values are increasing, decreasing or constant. There is a sense of maintaining order of function values as independent variable changes. These aspects are pictorially evident on the graph of a function. In a general case, a function may or may not maintain its order of change in its domain i.e. in the overall context. However, we can always identify monotonic behavior in an appropriately chosen subset of domain – unless it is a point function or a singleton.

Consider the graph of sine function. As a whole, the function is not monotonic as the order of the function is not preserved over the domain of the function, which is “R”. However, if we consider an interval, say, between “0” and “π/2”, then we find that function keeps increasing with the increasing independent variable. Therefore, sine function is monotonic in this interval.

On the other hand, a linear polynomial function represents a straight line, which maintains its monotonic nature through out its domain. The monotonic nature of a function, therefore, is investigated in a suitable interval, which is either domain or its subset. We shall refer this interval as X to illustrate the concept in this module. From the point of view of monotonic behavior, we classify function in following categories :

1: Constant function : Function values does not change as independent variable varies.

$$\text{If}{x}_1<x_2\text{then}f\left(x_1\right)=f\left(x_2\right),\text{for all}{x}_1,x_2\in X.$$

2: Strictly increasing: Function value change as independent variable varies in accordance with following condition :

$$\text{If}{x}_1<x_2\text{then}f\left(x_1\right)<f\left(x_2\right),\text{for all}{x}_1,x_2\in X.$$

3: Non-decreasing or increasing : Function value change as independent variable varies in accordance with following condition :

$$\text{If}{x}_1<x_2\text{then}f\left(x_1\right)\le f\left(x_2\right),\text{for all}{x}_1,x_2\in X.$$

4: Strictly decreasing: Function value change as independent variable varies in accordance with following condition :

$$\text{If}{x}_1<x_2\text{then}f\left(x_1\right)>f\left(x_2\right),\text{for all}{x}_1,x_2\in X.$$

5: Non-increasing or decreasing : Function value changes as independent variable varies in accordance with following condition :

$$\text{If}{x}_1<x_2\text{then}f\left(x_1\right)\ge f\left(x_2\right),\text{for all}{x}_1,x_2\in X.$$

There is one ambiguity in the definition of classification presented above. According to the definition, a constant function is an increasing, decreasing or both kinds of function. Clearly, this interpretation is wrong and is an exception. An increasing or non-decreasing class actually captures the notion of an overall increasing function, which is intermittently constant and thereby distinguishes this class from strictly increasing order. Similarly, a decreasing or non-increasing class actually captures the notion of an overall decreasing function, which is intermittently constant and thereby distinguishes this class from strictly decreasing order.

Note : It may confound clarity, but we should know that there is another classification. In this classification (i) "strictly increasing" is known simply as "increasing", (ii) "strictly decreasing" is known simply as "decreasing", (iii) "increasing" is known as "monotonically increasing" and (iv) "decreasing" is known as "monotonically decreasing". Clearly, this classification is not the same as what is given here. The best way to deal with this situation is to ignore this confusion and be explicit in what we mean. Saying "strictly increasing" for example ensures that equality of function values is not allowed. Similarly, saying "non-decreasing" ensures that function values do not decrease. We shall try to adhere to this explicit classification to the extent possible.