Continuity property

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In mathematics, the continuity property may be presented as follows.

Suppose that f : [a, b] → R is a continuous function. Then the image f([a, b]) is a closed bounded interval.

The theorem is the combination of the intermediate value theorem and the extreme value theorem, comprising the three assertions:

  1. The image f([a, b]) is an interval.
  2. This image is bounded.
  3. This image interval is closed, so f attains both its bounds.

The first assertion is broadly the intermediate value theorem; the latter two are equivalent to the extreme value theorem.

Contents

[edit] Proof of assertion 1

See: Intermediate value theorem#Proof

[edit] Proof of assertion 2

See: Extreme value theorem#Proof of the boundedness theorem

[edit] Proof of assertion 3

See: Extreme value theorem#Proof of the extreme value theorem

[edit] Caveats

It is important to note that this theorem only applies to continuous real functions. It does not apply to functions with function domain the rational numbers. As the rationals do not satisfy the least upper bound axiom, they are not complete.

To illustrate, this consider

f: [0,2] \cap \mathbb{Q} \to \mathbb{R}

x \mapsto e^{( - (x - \sqrt{2})^2 )}

f would obtain its maximum value at \sqrt{2} but this is not in the set.

If f is not continuous consider as a counterexample

f: [0,1] \to \mathbb{R}

x \mapsto \begin{cases} \frac{1}{x} & \frac{1}{x} \in \mathbb{Z} \\ 0 & \mbox{otherwise} \end{cases}

This is unbounded, but [0,1] is bounded.

Further, one should carefully note that the set must be closed, otherwise the maximum and minimum values might not be obtained.

[edit] References

  • Binmore, K. G. Mathematical Analysis: A Straightforward Approach, Cambridge University Press, (1982). ISBN 0521288827.