Composite of increasing functions is increasing

Statement

Statement for two functions

Suppose $f$ and $g$ are both functions of one variable that are increasing functions on their respective domains. Consider the composite of two functions $f\circ g$. This is also an increasing function on its domain.

Note that the statement makes no assumptions about the continuity or differentiability of the functions or even the nature of their domains. In fact, we do not even require that the domains and ranges be subsets of the real numbers, but only require that they be totally ordered sets so that the notion of increasing makes sense.

Statement for multiple functions

Fill this in later

Related facts

• Composite of two decreasing functions is increasing
• Composite of one-one functions is one-one

Proof

Proof for two functions

Given: $f$ and $g$ are increasing functions. $x_{1}<x_{2}$ are both in the domain of the composite function $f\circ g$.

To prove: $(f\circ g)(x_{1})<(f\circ g)(x_{2})$.

Proof:

Step no.	Assertion	Given data used	Previous steps used	Explanation
1	$g(x_{1})<g(x_{2})$	$g$ is increasing $x_{1}<x_{2}$		apply definition of increasing
2	$f(g(x_{1}))<f(g(x_{2}))$	$f$ is increasing	Step (1)	apply definition of increasing to inputs $g(x_{1}),g(x_{2})$, use Step (1).
3	$(f\circ g)(x_{1})<(f\circ g)(x_{2})$		Step (2)	Just rewrite Step (2) in terms of composite function, using the definition of composite function.