3. (Lecture 13) Let f : R+R be and let Xo E R. Suppose that f(x) > 0. (a) Assume that f is continuous at xo. Prove that

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3 Lecture 13 Let F R R Be And Let Xo E R Suppose That F X 0 A Assume That F Is Continuous At Xo Prove That 1 (60.32 KiB) Viewed 28 times

3. (Lecture 13) Let f : R+R be and let Xo E R. Suppose that f(x) > 0. (a) Assume that f is continuous at xo. Prove that there is an interval (a,b) CR with Xo E (a,b) such that for all x E (a,b) we have f(x) > 0. (b) Forget about the assumption in part (a). Prove that if, for all intervals (a,b) containing xo, there is an x E (a,b) such that f(x) < 0, then f is not continuous at xo.