Uniform Continuity: Polynomial of Degree 1 - What is \delta?

[juaninf]

hi everyone

I was reading one example about Uniform continuity, say that the polynomials, of degree less than or equal that 1 are Uniform continuity, my question is, for example in the case polynomial of degree equal to one Which is $$\delta$$, that the Uniform continuity condition satisfies.

thanks by you attention;

 

[snipez90]

Well we can do better and say that a polynomial on the reals is uniformly continuous if and only if the degree of the polynomial is < 2. The reverse implication is basically the general proof of what you're asking about.

In the case of a degree 1 polynomial, it's pretty easy. The polynomial is just a linear function defined by f(x) = ax + b. Given $\epsilon > 0$ you need to find a $\delta > 0$ for which $|x-y| < \delta$ implies $|f(x)-f(y)| < \epsilon$ for any real numbers x and y. If you're familiar with epsilon-delta proofs this should be easy.

 

[Anthony]

You need to talk about domains when you speak of uniform continuity. For instance, if X is compact, then any continuous function on X is necessarily uniformly continuous.