MATH 1271 Lecture Notes - Lecture 10: Intermediate Value Theorem
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We say f is continuous on (a,b) if f is continuous at every x in (a,b) We say f is continuous on the closed interval [a,b] if f is continuous on (a,b) and lim x->a+ f(x)=f(a) (right continuous) and lim x->b^- f(x)=f(b) (left continuous) Visual example f is continuous on (-infinity,-2)u(-2,-1)u(-1,1)u(1,3)u(3,infinity) f is right continuous at x=1 (and where f is continuous) f is left continuous at x=-1, x=3 (and where f is continuous) f is continuous on the closed interval [1,3] If f is continuous on the closed interval [a,b] and n is a number between f(a), f(b) and f(a)=/=f(b) hypothesis conditions to use theorem. Then there exists a c in (a,b) such that f(c)=n. Truth and falsehood of hypothesis and conclusion in a theorem. Show there exists a number whose cube is one more than the number itself.
