Proving Continuity of f(x) = x Using Deltas and Epsilons

[Mark44]

If delta < epsilon/2?

Yes.

Here's the proof that f(x) = 2x is continuous at (2, 4).
Let epsilon > 0 be chosen.

|f(x) - 4| < epsilon
==> |2x - 4| < epsilon
==> 2|x - 2| < epsilon
==> |x - 2| < epsilon/2

Take delta = epsilon/2

Since each of the steps above is reversible, if |x - 2| < delta, then |2x - 4| < epsilon, as required.

This example and your other example (f(x) = x) are very simple, since both functions represent straight lines. As soon as you increase the complexity to nonlinear functions, the proofs get quite a bit trickier.

 

[Moogie]

Hi

I don't think I'll go on to trickier problems :)

I just want to learn a bit of the proof behind some of the calculus I am learning as I feel a bit uncomfortable applying things without knowing where they come from. If I understand how the basic examples are done, I can feel comfortable the more complex examples can be prooved without knowing how to do them myself.

I'm from the UK and even in the calculus books i have for school/college level (up to 18 years old) they don't mention really touch on the theory of limits; they give an intuitive idea of what a limit is and then just dive straight in with differentiation and integration. The books I have now are american and have quite a different approach. Do you study this sort of thing at school/college in america? It seems in the UK you don't do it until university.

 

[Moogie]

If its not too much trouble, could i kindly ask you to show mw how to write out a quick proof that f(x) = x is continuous then I can write it up and put it with my notes so I can refer to it this time next week when I've forgotten it again.

I appreciate you have already spent more than enough time helping me on this question; your help it truly valued

 

[Mark44]

Here's the proof that f(x) = x is continuous at (a, f(a)) = (a, a).
Let epsilon > 0 be chosen.

|f(x) - f(a)| < epsilon
==> |x - a| < epsilon

Take delta = epsilon

Since each of the steps above is reversible, if |x - a| < delta, then |f(x) - f(a)| = |x - a| < epsilon, as required.

 

[Mark44]

Hi

I don't think I'll go on to trickier problems :)

I just want to learn a bit of the proof behind some of the calculus I am learning as I feel a bit uncomfortable applying things without knowing where they come from. If I understand how the basic examples are done, I can feel comfortable the more complex examples can be prooved without knowing how to do them myself.

I'm from the UK and even in the calculus books i have for school/college level (up to 18 years old) they don't mention really touch on the theory of limits; they give an intuitive idea of what a limit is and then just dive straight in with differentiation and integration. The books I have now are american and have quite a different approach. Do you study this sort of thing at school/college in america? It seems in the UK you don't do it until university.

I have been out of college math teaching for about 13 years now, so I haven't really been following calculus textbooks much. Different books seemed to take different approaches with regard to how rigorously limits were presented. I don't believe that high school texts do much in the way of epsilon-delta presentations of limits, but then I haven't seen any high school texts for a good long while.

 

[Moogie]

Thank-you. You've been incredibly helpful.