Suppose f and g are each defined on an open interval I, aI and aC(f)C(g). If f(a

Question

Suppose f and g are each defined on an open interval I, aI and aC(f)C(g). If f(a)>g(a), then there is an open interval J such that f(x)>g(x) for all xJ.

Explanation / Answer

We know that f and g are both continuous at a Now we know that f(a)>g(a) Say h(x) = f(x)-g(x) Now h(a) > 0 And h is continuous at a Hence there is a interval around a where h(x)>0 Hence there is a interval where f(x)>g(x) For interval part interval around a where h(x)>0 Use the epsilon delta definition of continuity, We know that h(a) > 0 and h is continuous at a by def, for all epsilon > 0 there exists a delta > 0 st |h(x) - h(a)|

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