# Definition:Pointwise Convergence

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## Definition

Let $\sequence {f_n}$ be a sequence of real functions defined on $D \subseteq \R$.

Suppose that:

- $\ds \forall x \in D: \lim_{n \mathop \to \infty} \map {f_n} x = \map f x$

That is:

- $\forall x \in D: \forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n > N: \size {\map {f_n} x - \map f x} < \epsilon$

Then **$\sequence {f_n}$ converges to $f$ pointwise on $D$ as $n \to \infty$**.

(See the definition of convergence of a sequence).

## Also defined as

Some sources insist that $N \in \N$ but this is unnecessary and makes proofs more cumbersome.

## Comment

Note that this definition of convergence of a function is weaker than that for uniform convergence, in which, given $\epsilon > 0$, it is necessary to specify a value of $N$ which holds for *all* points in the domain of the function.

In pointwise convergence, you need to specify a value of $N$ given $\epsilon$ for each individual point. That value of $N$ is allowed to be different for each $x \in D$.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $8.2$: Definition and examples: Definition $8.2.1$ - 1992: Larry C. Andrews:
*Special Functions of Mathematics for Engineers*(2nd ed.) ... (previous) ... (next): $\S 1.3$: Infinite series of functions - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**convergence of functions** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**pointwise convergence**