Final answer:
A sequence of functions f_n(x) = x^n on [0,1] converges pointwise to a function that is 0 on [0,1) and 1 at x=1, but this convergence is not uniform.
Step-by-step explanation:
An example of a sequence of continuous functions converging to a continuous function where the convergence is not uniform can be constructed using the functions fn(x) = xn on the interval [0,1].
Each function fn is continuous on [0,1], and the limit as n approaches infinity is the function f(x) which equals 0 for 0 ≤ x < 1 and 1 for x = 1.
Though each individual function and the limit are continuous, the convergence is not uniform because for any natural number n, no matter how large, there will always be points close to x = 1 where the difference between fn(x) and the limit function f(x) is significant.