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November 23

Real Analysis: Continuous Polynomials

I am trying to prove that polynomials are continuous using induction and the delta/epsilon definition. I am a little confused on one area of the k+1 part of the induction process.

How do I show that |x^k+1 - c^k+1| is continuous in order to obtain an epsilon? I know the product of 2 continuous function is continuous, should I implement that?

After I show that it is continuous, do I take epsilon= epsilon/2*|a| or =(epsilon/2*|a|)^-(k+1))? I need to get |a||x^k+1 - c^k+1| + epsilon/2 < epsilon.

If you can use that products are continuous, then it follows that powers are too; so, x^k is because x is, every polynomial is a sum of such functions, so, again, continuous. Another way to see this, quickly, is that real functions are continuous iff they preserve limits of sequences, you know that you can taking powers commutes with limits, ergo, x^k is continuous. If you require an epsilon-delta proof: follow the proof showing f * g is cont. if f and g are, substituting x for f and x^k for g; using induction on k, you get the result (or use it for hints to get the result yourself).

If you're looking for something a little less textbookish, but interesting to try: you can use oscillations and the binomial theorem with induction to prove it if you can show they are continuous at 0; or the fact that differentiable functions must be continuous, induction, and that ((x + h)^k - x^k)/h gets rid of k-power terms to show that they are all diff, hence cont.Phoenixia1177 (talk) 22:26, 23 November 2014 (UTC)[reply]