Definition:Pointwise Convergence

Definition

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

Suppose that $\displaystyle \forall x \in D: \lim_{n \to \infty} f_n \left({x}\right) = f \left({x}\right)$.

That is, $\forall x \in D: \forall \epsilon > 0: \exists N \in \R: \forall n > N: \left|{f_n \left({x}\right) - f \left({x}\right)}\right| < \epsilon$.

Then $\left \langle {f_n} \right \rangle$ converges to $f$ pointwise on $D$ as $n \to \infty$.

(See the definition of convergence of a sequence).

Note

Some sources insist that $N \in \N$ but this is not strictly necessary and can make 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 $x \in D$.