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I must understand this proof of continuity. I understand the basics of continuity and algebra of continuity of limits. So I add the picture of the proof such that it is in the book of Real Analysis.I would really appreciate if someone could explain it to me as simple as possible. I'm struggling to understand it, What do $\min{[f,g]}$ and $\max{[f,g]}$ mean? Thank you.

Malu
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3 Answers3

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$\max(f,g)=\begin{cases}f \text{ if }f\ge g \\\\g\text{ otherwise }\end{cases}$

$\min(f,g)=\begin{cases}f \text{ if }f\le g \\\\g\text{ otherwise }\end{cases}$

J. W. Tanner
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For every $x$ in a neighborhood of $a$, you define

$$\operatorname{max}\{f,g\}(x):=\operatorname{max}\{f(x), g(x)\}$$ Similarly for $\operatorname{min}\{f,g\}$.

Maryam
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  • For every x in a neighborhood of a, ok this is useful. Related to $x\rightarrow a$. We re talking about intervals. – Malu May 24 '20 at 21:33
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These two functions are defined as follows: whenever $f(x) \geq g(x)$, we have $$\max \lbrace f,g \rbrace (x):= f(x) $$ while, whenever $g(x) \geq f(x)$, we have: $$ \max \lbrace f,g \rbrace (x):= g(x) $$ The function $\min \lbrace f,g \rbrace$ is defined analogously. Regarding the equations in the proof, see also: How to show that $\max(f,g) =(f+g+|f-g|)/2$?.

Manuel Norman
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