## Theorem 8.2.3: Uniform Convergence preserves Continuity

If a sequence of functions

*f*defined on_{n}(x)*converges uniformly to a function***D***f(x)*, and if each*f*is continuous on_{n}(x)*, then the limit function***D***f(x)*is also continuous on*.***D**All ingredients will be needed, that *f _{n}* converges
uniformly and that each

*f*is continuous. We want to prove that

_{n}*f*is continuous on

*. Thus, we need to pick an*

**D***x*and show that

_{0}|f(x_{0}) - f(x)| < if |x_{0}- x| <

Let's start with an arbitrary * > 0*.
Because of uniform convergence we can find an *N* such that

|f_{n}(x) - f(x)| < /3 if n N

for all *x D*.
Because all

*f*are continuous, we can find in particular a

_{n}*> 0*such that

|f_{N}(x_{0}) - f_{N}(x)| < /3 if |x_{0}- x| <

But then we have:

|f(x_{0}) - f(x)| |f(x_{0}) - f_{N}(x_{0})| + |f_{N}(x_{0}) - f_{N}(x)| + |f_{N}(x) - f(x)|

/3 + /3 + /3 =

as long as *|x _{0} - x| < *.
But that means that

*f*is continuous at

*x*.

_{0}