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Uniform continuity

  1. Jul 22, 2010 #1
    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 [tex]\delta[/tex], that the Uniform continuity condition satisfies.

    thanks by you attention;
     
  2. jcsd
  3. Jul 23, 2010 #2
    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 [itex]\epsilon > 0[/itex] you need to find a [itex]\delta > 0[/itex] for which [itex]|x-y| < \delta[/itex] implies [itex]|f(x)-f(y)| < \epsilon[/itex] for any real numbers x and y. If you're familiar with epsilon-delta proofs this should be easy.
     
  4. Jul 23, 2010 #3
    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.
     
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