Define for . The sequence of real valued functions converges pointwise to a function if for every there exists such that for and for each we have .
If a sequence of functions converges pointwise to a limit , then can be found by
Pointwise convergence need not preserve continuity, for example define for
and observe that the limit for
and for we have
which means that may be written
This function is trivially not continuous, despite all the functions in the sequence being perfectly continuous over the domain. For a stronger form of convergence that preserves continuity, see uniform convergence.