# Calculus and Topology:

• The Intermediate Value Theorem
• The Extreme Value Theorem
• A Fixed Point Theorem

•

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),
then there is a real number c between a and  b with f (c) = v.

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.

• Topological Proof
• 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.

• The Extreme Value Theorem :
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].
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.

• Topological Proof
• 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  B³ 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.