Let F and g be continous function from R\' to R\'. Define F(x) = max[ f(x), g(x)

Question

Let F and g be continous function from R' to R'. Define F(x) = max[ f(x), g(x) ] for each x R'. Show that F is continous.

Explanation / Answer

FOLLOW THIS Given: f,g: R ? R continuous So,|x - c| < del ==> |f(x) - f(c)| < epi. (i) and |x - c| < del ==> |g(x) - g(c)| < epi. (ii) Now, Let h(x)=max {f(x),g(x)}. To Prove: h(x) is continuous. Proof: To show h(x) to be continuous it is enough to show that the following holds... for |x - c| < del ==> |h(x) - h(c)| < epi. (iii) Observe from (i) and (ii) that for |x - c| < del ==> |max{f(x),g(x)} - max{f(c),g(c)}| < epi (iv) Since f(x) and g(x) is continuous in (x - del, x + del) interval. Max{} is bounded. And hence (iv) is true. So by (iii) h(x) is continous.

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