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
Propositions: 1