Consider the one-dimensional heat equation $ u_{xx}=u_t$ with the boundary data
$$u(x, 0)=f(x), \quad u(0, t)=u(L, t)=0.$$
The standard method of solving this equation is by finding a candidate family of solutions, by means of separation of variables, and then checking that the candidate $u(x, t)$ indeed satisfies both the equation and the boundary data. Regarding showing that the expected solution has the asserted boundary values , Knapp in his Advanced real analysis book writes "This amounts to interchanging infinite sums with passages to the limit as certain variables tend to the boundary, and the following result can often handle that."
Proposition. Let $X$ be a set, let $Y$ be a metric space, let $A_n (x)$ be a sequence of complex-valued functions on $X$ such that $\sum_{n=1}^\infty|A_n (x)|$ converges uniformly, and let $B_n (y)$ be a sequence of complex-valued functions on $Y$ such that $|B_n (y)| \le1$ for all n and y and such that $\lim_{y\to y_0} B_n (y) = B_n (y_0 )$ for all $n$. Then
$$\lim_{y\to y_0}\sum_{n=1}^\infty A_n(x)B_n(y)=\sum_{n=1}^\infty A_n(x)B_n(y_0) $$ and the convergence is uniform in $x$ if, in addition to the above hypotheses, each $A_n (x)$ is bounded.
My question is why do we need to check, for instance, $\lim_{t\to 0}u(x, t)=f(x)$, rather than $u(x, 0)=f(x)$ by simply plugging in $t=0$.
Edit: I suspect that this is because Knapp wants the function $u$ be continuous over its domain of definition. (See the comments below). However Knapp writes "The conditions $u(0, t)=u(L, t)=0$ may be verified in the same way" by the above proposition which literally means that he uses the proposition in evaluation of $u$ at boundary points which to me doesn't sound right.
