As we move on the graph of a function from left to right (which corresponds to the increase in the argument $x$) if the graph constantly moves upwards, we say that the function is* increasing* (Figure 1(a)), and if the graph constantly moves downward, we say that the function is *decreasing* (Figure 1(b)).

**Definition:**Let $f(x)$ be defined on an interval $I$.

- We say $f(x)$ is
**increasing**on $I$, if for every pair of points $x_{1},x_{2}$ in $I$ satisfying the condition $x_{1}<x_{2}$, we have $f(x_{1})<f(x_{2})$. - We say $f(x)$ is
**decreasing**on $I$, if for every pair of points $x_{1},x_{2}$ in $I$ satisfying the condition $x_{1}<x_{2}$, we have $f(x_{1})>f(x_{2})$.

- Note that $I$ can be finite (or bounded) or infinite (unbounded).
- An interval on which the function is increasing is called an
**interval****of increase**of a function while an interval on which the function is decreasing is called an**interval of decrease**. - A function that is either increasing or decreasing on an interval is said to be
**monotonic**on the interval.

For example, $f(x)=x^{2}$ is decreasing on $(-\infty,0]$ and is increasing on $[0,\infty)$ (Figure 2(a)). The function $g(x)=x^{3}$ is increasing on $(-\infty,\infty)$ (Figure 2(b)).