Given real-valued continuous functions $f, g$, is the following (and why?) inequality true?
$$\max \{f + g \} \leq \max f + \max g$$
Can someone give me a proof? I suspect the min is the reverse inequality
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7$(f+g)(x)=f(x)+g(x)$ and $f(x)\leq \max f$, and $g(x)\leq \max g$. – Álvaro Lozano-Robledo Mar 10 '13 at 20:18
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Assuming the maxima *exist*, then yes, that is true. In general, you would have to use the supremum. – Zev Chonoles Mar 10 '13 at 20:22
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Oh so $\max(f + g)(x) \leq \max \max (f + g) \leq \max ( \max f + \max g ) = \max f + \max g$ – Lemon Mar 10 '13 at 20:22
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@sizz That's an interesting way of looking at it. – goblin GONE Mar 10 '13 at 20:23
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Remark: Use the following result. If $f$ has domain $X$, and if $a$ is a real number, and if for all $x \in X$ it holds that $f(x)\leq a$, then $\max f \leq a$. – goblin GONE Mar 10 '13 at 20:32
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possible duplicate of Suprema proof: prove $\sup(f+g) \le \sup f + \sup g$ – Mar 07 '14 at 15:37
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@goblin, A counter example of the Remark: $X = \mathbb{R}$ and $f(x) = 1 - e^{-x^2}$. $\sup f = 1$, but the maximum doesn't exist and $f(x) \leq 1$ for all $x$. – Jae Young Lee Apr 27 '20 at 20:28
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Basically, I'm curious about how to prove that $\mathrm{max} {f + g}$ exists under the existence of $\mathrm{max} f$ and $\mathrm{max} g$. – Jae Young Lee Apr 27 '20 at 20:29
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For all $x$, $$f(x)\le \max f(x)$$ and $$g(x)\le\max g(x).$$ Now add the two together.
MJD
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let $\max g(z):=g(z^*)$, $\max f(z):=f(z^*)$ then $$f(z)\le f(z^*)$$ $$g(z)\le g(z^*) $$ then $$\forall z : (f+g)(z)=f(z)+g(z)\le f(z^*)+g(z^*)$$ $$\max(f+g)(z)\le f(z^*)+g(z^*)=\max f(z)+\max g(z)$$
Martin Sleziak
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M.H
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1the only way to replace max $g(x)$ by $g(x^)$ is that g is continuous and the domain should be compact, for example if the domain is $\Bbb R$ and g=x you cant replace max g by $g(x^)$, that is the max is not necessarily an image. – i.a.m Mar 10 '13 at 20:52
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More generally we have:$$\sup_{x\in A} ( f(x)+g(x))\le \sup_{x\in A} \left( f(x)+\sup_{y\in A} g(y)\right)=\sup_{x\in A} f(x)+\sup_{y\in A} g(y)$$