6.10. Sequences of Continuous FunctionsSometimes it it difficult to access properties of a function f directly. An alternative idea is to approximate f by simpler functions which however requires that we only test properties which cooperate with the approximation method used. In this part we develop such a concept for continuity.
Consider:
The last example displays a certain shortcoming in pointwise convergence: Although all sequence members, i.e. the powers $\mathrm{X}}^{n$, are continuous on $[\mathrm{0,1}]$ this is
no longer true
i f is in fact discontinuous at 1 as $1\frac{1}{n}\to 1$, but
${f}_{n}(x)f(x)<\epsilon$
for all $n\ge {n}_{0}$ this $n}_{0$ however may vary from x to x. Thus we need a stronger notion of convergence to get it "continuityproof".
Consider:
The technique used in the proof above will be generalised to a test in [6.10.7]. But first we will show that uniform convergence respects continuity.
It is often difficult to see if a sequence of functions is uniformly convergent or not so that we would benefit from appropriate tests. We prove two of them, a kind of comparison test and the wellknown Cauchytest. The technical advantage of Cauchy's test is that we don't need to know the limit function, the disadvantage on the other hand that we won't get it.
As an example we consider the sequence $(\frac{n\mathrm{X}}{1+{n}^{2}{\mathrm{X}}^{2}})$. For any $0<c<1$ this sequence converges uniformly on $[c,1]$ to 0: The estimate
$\frac{nx}{1+{n}^{2}{x}^{2}}0\le \frac{n\cdot 1}{1+{n}^{2}{c}^{2}}$ for all $x\in [c,1]$
proves $(\frac{n}{1+{c}^{2}{n}^{2}})$ to be suitable for [6.10.7].
