Let $f(x)$ and $g(x)$ are decreasing function. Can we conclude that $f(x) + g(x)$ is also decreasing ? In general can we deduct that sum of monotonic functions is also monotonic and if functions are increasing or decreasing then sum of them is also increasing or decreasing ? If these statements are true , how we can prove it ?
Sum of two decreasing function
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Yes. Assume $f(x)$ and $g(x)$ are decreasing in $[a,b]$ and were defined in this interval. For every $s$ and $t$ that $a\leq s\leq t\leq b$ we have $$f(s)\leq f(t)\hspace{1cm};\hspace{1cm}g(s)\leq g(t)$$ so $$f(s)+g(s)\leq f(t)+g(t)$$
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0So sum of several increasing functions is increasing and sum of several decreasing functions is decreasing . Is it right ? – 2017-01-23
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0They must be define in same set. – 2017-01-23
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0Yes they are defined in same interval – 2017-01-23