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I have 2 functions: $f(x)$ and $g(x)$. Both of them range in $[0,1]$.

Is there some way to define a $h(x)$ that efficiently takes the greater one of $h(x)$ and $g(x)$, i.e. $h(x) = \max \{ f(x),g(x) \}$, but without using $\max$?

If there is no exact solution, an approximate $h(x)$ is also fine.

Jokester
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1 Answers1

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Since you ask for alternatives, note that $$\max\{s,t\}=\tfrac12s+\tfrac12t+\tfrac12|s-t|, $$ and that, if one also wants to avoid absolute values, $$\max\{s,t\}=\tfrac12s+\tfrac12t+\tfrac12\sqrt{t^2+s^2-2st}. $$ Both these formulas are actually less handy than the maximum you started from (the square-root formula being probably even less handy than the absolute-value one)...

Did
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  • Nice point! I agree that the $max$ is probably more handy. @Jokester And by the way, what do you mean for efficient? What do you want to do? – AndreaCassioli May 03 '14 at 11:07
  • Here is one possible application of the former formula: after proving that the Riemann integrable functions form a vector space over the reals, and proving that absolute value preserves Riemann integrability, it becomes easy to show that max preserves Riemann integrability. – Fengyang Wang Mar 05 '16 at 01:07