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# 12.1 Continuity

12.1   Definition (Continuity at a point.) Let be a real valued function such that . Let . We say that is continuous at if and only if

Remark: According to this definition, in order for to be continuous at we must have

and

The second condition is often not included in the definition of continuity, so this definition does not quite correspond to the usual definition.

Remark: The method we will usually use to show that a function is not continuous at a point , is to find a sequence in such that , but either diverges or converges to a value different from .

12.2   Definition (Continuity on a set.) Let be a real valued function such that domain , and let be a subset of domain. We say that is continuous on if is continuous at every point in . We say that is continuous if is continuous at every point of domain.

12.3   Example (, , and power functions are continuous.) We
proved in lemma 11.17 that a function is continuous at every point at which it is differentiable. (You should now check the proof of that lemma to see that we did prove this.) Hence , , , and (for ) are all continuous on their domains, and if , then is continuous on .

12.4   Example. Let

Then is not continuous at . For the sequence converges to , but .

Our limit rules all give rise to theorems about continuous functions.

12.5   Theorem (Properties of continuous functions.) Let be real valued functions with , and let . If and are continuous at and if is approachable from , then and are continuous at . If in addition, then is also continuous at .

Proof: Suppose and are continuous at , and is approachable from . Then

By the sum rule for limits (theorem 10.15) it follows that

Thus is continuous at . The proofs of the other parts of the theorem are similar.

12.6   Example (An everywhere discontinuous function.) Let be the example of a non-integrable function defined in equation (8.37). Then is not continuous at any point of . Recall

where is a subset of such that every subinterval of of positive length contains a point in and a point not in . Let .

Case 1.
If we can find a sequence of points in such that . Then

so is not continuous at .

Case 2.
If we can find a sequence of points in such that . Then

so is not continuous at .

12.7   Example. Let

I claim that is continuous. We know that is differentiable on , so is continuous at each point of . In example 10.13 we showed that so is also continuous at .

12.8   Example. Let

Then and are both continuous functions. Now

and hence is not continuous. The domain of contains just one point, and that point is not approachable from .

12.9   Theorem (Continuity of compositions.) Let be functions with domains contained in and let . Suppose that is continuous at and is continuous at . Then is continuous at , provided that is approachable from .

Proof: Suppose is continuous at and is continuous at , and is approachable from . Let be a sequence in such that . Then since is continuous at . Hence since is continuous at ; i.e.,

Hence is continuous at .

Next: 12.2 A Nowhere Differentiable Up: 12. Extreme Values of Previous: 12. Extreme Values of   Index
Ray Mayer 2007-09-07