Max Operation on Continuous Real Functions is Continuous

Theorem

Let $f: \R \to \R$ and $g: \R \to \R$ be real functions.

Let $f$ and $g$ be continuous at a point $a \in \R$.

Let $h: \R \to \R$ be the real function defined as:

$\map h x := \map \max {\map f x, \map g x}$

Then $h$ is continuous at $a$.

Proof

From Max Operation Representation on Real Numbers

$\max \set {x, y} = \dfrac 1 2 \paren {x + y + \size {x - y} }$

Hence:

$\max \set {\map f x, \map g x} = \dfrac 1 2 \paren {\map f x + \map g x + \size {\map f x - \map g x} }$

From Difference Rule for Continuous Real Functions:

$\map f x - \map g x$ is continuous at $a$.

From Absolute Value of Continuous Real Function is Continuous:

$\size {\map f x - \map g x}$ is continuous at $a$.

From Sum Rule for Continuous Real Functions:

$\map f x + \map g x$ is continuous at $a$

and hence:

$\map f x + \map g x + \size {\map f x - \map g x}$ is continuous at $a$

From Multiple Rule for Continuous Real Functions:

$\dfrac 1 2 \paren {\map f x + \map g x + \size {\map f x - \map g x} }$ is continuous at $a$.

$\blacksquare$

Also see

• Min Operation on Continuous Real Functions is Continuous

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: Exercise $1.5: 18 \ \text {(b)}$