Composite of increasing functions is increasing
Statement for two functions
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
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Related facts about composites of functions
Related facts about increasing functions
- Inverse of increasing function is increasing
- Increasing functions form a cone in a vector space
- Product of increasing functions need not be increasing
Proof for two functions
Given: and are increasing functions. are both in the domain of the composite function .
To prove: .
|Step no.||Assertion||Given data used||Previous steps used||Explanation|
|1|| is increasing
||apply definition of increasing|
|2||is increasing||Step (1)||apply definition of increasing to inputs , use Step (1).|
|3||Step (2)||Just rewrite Step (2) in terms of composite function, using the definition of composite function.|
Compatibility with chain rule for differentiation
In case both the functions and are differentiable, then we can check that the statement is compatible with the chain rule for differentiation. In fact, the chain rule for differentiation can also be used to furnish an alternative proof in this case, though we have to deal carefully with the case of zero derivative.
- If both and are positive everywhere on their domain, so is .
- If both and are nonnegative everywhere on their domain, so is .
These almost prove that a composite of increasing differentiable functions is increasing.