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Definition

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 .

Properties

If a sequence of functions converges pointwise to a limit , then can be found by

Continuity

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.

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