## Real Analysis – Limits and Continuity V

The condition is somewhat hard to get into our heads as neophytes. On top of that the similarity of the definition for limit and continuity can increase the confusion and to try to counter those frequent turn of events the first part of this post will try to clarify the condition by means of examples.

** — for Continuity —**

First we’ll start things off with something really simple.

Let which is obviously continuous.

The gist of the the reasoning is that we want to show that no matter the that is chosen at first it is always possible to find an that satisfies Heine’s criterion for continuity.

Getting back to our function it is . Here so

Which is trivially true since by assumption. Hence any value of will satisfy Heine’s criterion for continuity and is continuous at .

Since we never made any assumption about other than we conclude that is continuous in all points of its domain.

Let us now look at . Again we’ll look at continuity for point ():

The last expression is just we want at this stage since want to have something of the form (the first part of the criterion).

If we let it is and this completes our proof that is continuous at point .

And again since we never made any assumption about other than we conclude that is continuous in all points of its domain.

Now we let and will see if is continuous at .

Hence if we let it is and is continuous at .

As a final example of Heine’s criterion of continuity we’ll look into .

Since we want something like the last expression isn’t very useful to us.

In this case we’ll take an alternative approach which nevertheless works and has exactly the same spirit of what we’ve using so far.

Please look at every step I make with a critical eye and see if you can really understand what’s going on with this deduction.

Since we know that at some point will be in the first quadrant. Thus

Where the last inequality follows by hypothesis.

That is to say that if we let it is which is the epsilon delta definition of continuity.

** — for Limits —**

After looking into some simple proofs for continuity we’ll take a look at for limits.

The procedure is the same, but we’ll state it explicitly so that people can see it in action.

Let . We want to show that it is .

Which is trivially true for any value of , hence can be any positive real number.

Let . We want to show that it is .

With we satisfy the for limit.

As a final example let us look at the modified Dirichlet function that was introduced at this post.

At that post it was proved that for didn’t exist and it was promised that in a later date I’d show that using the epsilon delta condition.

Since we now know what the epsilon delta condition is and already have some experience with it will tackle this somewhat more abstruse problem.

Since or we have two cases to look at.

In the first case it is which is trivially valid, hence can be any real positive number.

In the second case it is . Hence letting gets the job done.

Since we proved that the conclusion is that the modified Dirichlet function that was presented is only continuous at .

As was said previously, they don’t make local concepts more local than that.

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