Here we will show that having two sequences of functions, { f_k} that converges to some f on a set M and {g_k} that converges to some g on a set N (and assuming that all g_k map N into M), we cannot claim that { f_k(g_k)} converges to f (g), not even if the functions involved are continuous.

Consider the functions f_k(x) = arctan(k⋅x), we know that they form a convergent sequence and

Denote this limit by f.

Consider also the functions g_k(x) = x/k, it is easy to see that they converge to the constant function g(x) = 0 on the whole real line.

What can we say about compositions? For all k we have f_k(g_k(x)) = arctan(x), so { f_k(g_k)} is a constant sequence of functions that converges to the function arctan(x). On the other hand, f (g) is the constant function f (g(x)) = f (0) = 0, so { f_k(g_k)} definitely does not converge to f (g).