Real Analysis/Pointwise Convergence

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Real Analysis
Pointwise Convergence

Let be a sequence of functions defined on a common domain . Then we say that converges pointwise to a function if for each the numerical sequence converges to . More precisely speaking:

For any  and for any , there exists an N such that for any n>N, 

An example:

The function

converges to the function

This shows that a sequence of continuous functions can pointwise converge to a discontinuous function.