The Intermediate Value Theorem :
If f is a continuous function and a < b and v are real numbers with v between f(a) and f(b),Comment: A traditional analytic proof of this theorem uses either the least upper bound property of real numbers or related analytic tools using sequences and the analytic definitions of continuity and limits.
then there is a real number c between a and b with f (c) = v.
Topological Facts: A closed interval is a connected set. If X is a connected set of real numbers with elements d<e, then X contains [d,e]. If f is a continuous function and X is a connected set, then f(X) is also connected. Proof: [a,b] is a connected set, and therefore f ([a,b]) is a connected set containing f(a) and f(b). Thus the interval I determined by f (a) and f (b) is contained in f ([a,b]) . But the assumption is that v is an element of the interval I, and therefore v is a member of f ([a,b]), i.e., there is a real number c between a and b with f (c) = v. EOP.
Comment: A traditional analytic proof of this theorem uses either the least upper bound property of real numbers or related analytic tools using sequences and the analytic definitions of continuity and limits.If f is a continuous function and a < b, then there is a real number c between a and b with f (c) ³ f(x) for all x in the interval [a,b].
Topological Facts: A closed and bounded subset of the real numbers is a compact set. If X is a compact set of real numbers, then X contains both closed and bounded. A compact set of real numbers contains a largest element. If f is a continuous function and X is a compact set, then f(X) is also compact. Proof: [a,b] is a compact set, and therefore f ([a,b]) is a compact set. Thus f ([a,b]) is a closed and bounded set. So f ([a,b]) contains an element B with B³y for all y in f ([a,b]) .
i.e., there is a real number c between a and b with f (c) ³ f(x) for all x in the interval [a,b]. EOP.
A Fixed Point Theorem:
If f is a continuous function from [0,1] into [0,1] then there is some number a with f(a) = a.