Final answer:
The question seeks an example of pointwise but not uniform convergence of continuous functions on (0,1). The sequence x^n is continuous and converges pointwise to 0 on (0,1) but does not converge uniformly, as for any n, x close to 1 can be found where x^n is not close to 0.
Step-by-step explanation:
The question asks for an example of a sequence of continuous functions on the interval (0,1) that converges pointwise to a continuous function, but where this convergence is not uniform.
A classic example of such a sequence is given by fn(x) = xn. Each function in this sequence is continuous on (0,1), and as n approaches infinity, fn(x) converges pointwise to a function f(x) which is 0 for all x in (0,1) and 1 at x=1. The function f(x) can be extended to be continuous on (0,1) by defining it to be 0 for all x in (0,1).
However, the convergence is not uniform because, for any given n, no matter how large, there exists an x close to 1 for which fn(x) is significantly different from 0.
To show this formally, one can use the definition of uniform convergence, which requires that for any ε > 0, there exists an N such that for all n > N and all x in (0,1), |fn(x) - f(x)| < ε. For the given sequence, one can always find an x close enough to 1 such that this condition fails, therefore, the convergence cannot be uniform on the interval (0,1).