Daily Definition 10/19

On Thursday 10/19 we went over uniform convergence and the Sequential Characterization, both of which I will address in this post.

Definition: Given f: A \rightarrow \mathbb{R} , \forall \epsilon >0, \exists \delta>0 (depending only on \epsilon) so that \forall c \in A it holds that whenever x\in A, |x-c| < \delta, |f(x)-f(c)| <\epsilon.

In class we went over three functions, two of which I will touch on here.

The first is the function g(x) = x^2 , A =[0,\infty) and the second is the function h(x) = \sqrt(x), also with the domain A=[0,\infty). With our math background, it is obvious that both these functions are continuous because they contain no holes, gaps, jumps, etc. They are smooth curves. However, the function g(x) = x^2 is not uniformly continuous while h(x) = \sqrt(x) is uniformity continous. So what is the difference between the definition of continuity versus uniform continuity?

Well, for either of these functions to be continuous on their domain A, for every individual point c \in g(x) (or c\in h(x)), given an \epsilon >0 we simply need to find a \delta>0 such that if |x-c| < \delta it follows that |f(x)-f(c)|< \epsilon. Now uniform continuity says that given \epsilon>0 We can find one \delta>0 where for any c\in A, it holds that |x-c| < \delta, |f(x)-f(c)| <\epsilon. For this reason, g(x) = x^2 is not uniformity continuous. With a set \epsilon window, as x \in A increases, the \delta window decreases, so it is impossible to find a \delta window that works for any point in the domain.

 

For completeness, I will state the Sequential Charaterization: a function f is uniformly continuous on a domain A, if and only if, \forall (x_n),(y_n) \subset A, |x_n-y_n| \rightarrow 0 implies $latex|f(x_n)-f(y_n) \rightarrow 0$. (This can be used to prove g(x) is not uniformly continuous and to prove that h(x) is uniformly continuous).

 

Posted on by Brittney D'Oleo
Categories: Definitions

One Response

  1. Jeremy LeCrone says:
    October 26, 2017 at 12:43 pm

    Thank you for the exploration of definitions, Brittney. I am confused by the notation c \in g(x), however, because g(x) is a value in R, not a set, and it is a value in the range of the function g. Then, you go on to compare the distance from c to x, which is in the domain of the function g.

    Also, you should note carefully that the \delta windows decreasing is not quite enough to confirm lack of uniform continuity, you need to know that it continues decreasing until the limit of the delta value is in fact zero (so that there is no positive lower bound on the set of admissible deltas)…

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