Min Operation on Continuous Real Functions is Continuous
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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:
- $\map h x := \map \min {\map f x, \map g x}$
Then $h$ is continuous at $a$.
Proof
From Min Operation Representation on Real Numbers
- $\min \set{x, y} = \dfrac 1 2 \paren {x + y - \size {x - y} }$
Hence:
- $\min \set {\map f x, \map g x} = \dfrac 1 2 \paren {\map f x + \map g x - \size {\map f x - \map g x} }$
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 from Difference Rule for Continuous Real Functions again:
- $\map f x + \map g x - \size {\map f x - \map g x}$ 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
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: Exercise $1.5: 18 \ \text {(b)}$