**The
Intermediate Value Theorem :**

Iffis a continuous function anda < bandvare real numbers withvbetweenf(a) andf(b),then there is a real numbercbetweenaandbwithf(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 alsoconnected.Proof:[ a,b] is a connected set, and thereforef([a,b]) is a connected set containingf(a) andf(b).Thus the interval I determined by f(a) andf(b) is contained inf([a,b]) .But the assumption is that vis an element of the interval I, and thereforevis a member off([a,b]),i.e.,there is a real numbercbetweenaandbwithf(c) =v.EOP.

Iffis a continuous function anda < b,then there is a real numbercbetweenaandbwithf(c) ³f(x) for allxin 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 alsocompact.Proof:[ a,b] is a compact set, and thereforef([a,b]) is a compact set.Thus f([a,b]) is a closed and bounded set. Sof([a,b]) contains an element B with B³yfor all y in f([a,b]) .

i.e.,there is a real numbercbetweenaandbwithf(c) ³f(x) for allxin the interval [a,b].EOP.

A Fixed Point Theorem:

Iffis a continuous function from [0,1] into [0,1] then there is some numberawithf(a) =a.