Section: Uniform Convergence of Functions

The concept of convergence of real sequences can be carried over to the concept of convergence of functions. This is possible if we have infinitely many functions $f_n:D\to\mathbb R,$ (i.e. for $n=0,1,2,\ldots$) which: * all have the same domain $D$ * and at every point $x\in D$ the real sequence $(f_n(x))_{n\in\mathbb N}$ converges to a limit to a $\lim_{n\to\infty} f_n(x)=f(x).$

Such a pointwise convergence to the function $f$ (i.e. at the point $x\in D$) is not, in general, sufficient to be able to predict the properties of the limit function $f,$ if we known the properties of the infinitely many given functions $f_n.$ Concluding the properties of $f$ based on the known properties of $f_n$ becomes, however, possible, if the $f_n$ converge "similarly quickly" and at "every point" $x\in D.$ This so called uniform convergence is what this section of BookofProofs will be about. It will contain a rigorous definition as well as the rules, which other properties exactly of $f$ can be concluded based on those of $f_n$ if uniform convergence is in place.

Examples: 1

  1. Definition: Pointwise and Uniformly Convergent Sequences of Functions
  2. Proposition: Only the Uniform Convergence Preserves Continuity
  3. Definition: Supremum Norm for Functions
  4. Proposition: Supremum Norm and Uniform Convergence
  5. Proposition: Uniform Convergence Criterion of Cauchy
  6. Proposition: Uniform Convergence Criterion of Weierstrass for Infinite Series
  7. Proposition: Calculations with Uniformly Convergent Functions

Propositions: 1

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983