Real Analysis/Pointwise Convergence

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Real Analysis
Pointwise Convergence

Let f_n(x)\, be a sequence of functions defined on a common domain D\subseteq\mathbb{R}\,. Then we say that f_n(x)\, converges pointwise to a function f(x)\, if for each x\in D\, the numerical sequence f_n(x)\, converges to f(x)\,. More preciselly speaking:

For any x\in D\, and for any \varepsilon>0\,, there exists an N such that for any n>N, \left|f_n(x)-f(x)\right|<\varepsilon

An example:

The function

f_n(x) = \frac{x^n}{1+x^n} converges to the function

f(x) = \left\{ \begin{array}{ll} 1 & \text{if } |x| > 1\\ \frac{1}{2} & \text{if } x = 1\\ 0 & \text{if } |x| < 1 \\ \end{array} \right.

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