Definition:Pointwise Convergence

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

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

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 $$x \in D$$.