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Math and science::Analysis::Tao::09. Continuous functions on R

The maximum principle

Continuous functions whose domain is a closed enjoy two useful properties:

  • the maximum principle
  • the intermediate value theorem

This card covers the first. If a function is continuous and it's domain is a closed interval, then it has a maximum value (one or more values of the domain map to a maximum).

Bounded functions

Let XR and let f:XR be a function.

We say that f is bounded from above iff [...].

We say that f is bounded from below iff [...].

We say that f is bounded iff it is both bounded from above and bounded from below.

[...] are bounded

Let a<b be real numbers. Let f:[a,b]R be a function [...] on [a,b]. Then f is a bounded function.

Obtains maximum/minimum

Let f:XR be a function, and x0X.

We say that f obtains its maximum at x0 iff f(x0)f(x) for all xX.

We say that f obtains its minimum at x0 iff f(x0)f(x) for all xX.

Maximum principle

Let a<b be real numbers. Let f:[a,b]R be a function continuous on [a,b].

Then there is an xmax[a,b] such that f(xmax)f(x) for all x[a,b] (f obtains its maximum at xmax).

Similarly, there is an xmin[a,b] such that f(xmin)f(x) for all x[a,b] (f obtains its minimum at xmin).