If a sequence of functions fn(x) defined on D converges uniformly to a function f(x), and if each fn(x) is continuous on D, then the limit function f(x) is also continuous on D.Context
Theorem 8.2.3: Uniform Convergence preserves Continuity |
If a sequence of functions fn(x) defined on D converges uniformly to a function f(x), and if each fn(x) is continuous on D, then the limit function f(x) is also continuous on D. |
All ingredients will be needed, that fn converges uniformly and that each fn is continuous. We want to prove that f is continuous on D. Thus, we need to pick an x0 and show that
|f(x0) - f(x)| <if |x0 - x| <
Let's start with an arbitrary > 0.
Because of uniform convergence we can find an N such that
|fn(x) - f(x)| <N
for all x 
D.
Because all fn are continuous, we can find in particular
a > 0 such that
|fN(x0) - fN(x)| <
But then we have:
|f(x0) - f(x)||f(x0) - fN(x0)| + |fN(x0) - fN(x)| + |fN(x) - f(x)|
as long as |x0 - x| < .
But that means that f is continuous at x0.
