Q:

Pointwise convergence is enough for continuity of operators limit, but not of functions

In Functional Analysis, we learned that pointwise convergence for a linear and continuous map (operator) between Banach spaces is enough to guarantee that the limit will be continuous.
However, in simple calculus we also learned that pointwise convergence was not enough for guaranteeing continuity of the limit of continuous functions. We would need to have uniform convergence( convergence in norm)
What's the main ingredient that allows to conciliate both pieces of information, and why?
Is it because in the first we're restricting ourselves to just linear maps?

A:

Let $f_n:\mathbb R \to \mathbb R$ be linear maps conveging pointwise to some function $f$. . Then $f_n(x)=c_nx$ for some constants $c_n$. Since $f(1)=\lim f_n(1)=\lim c_n$ it follows that $f(x)=\lim c_nx =f(1)x$, so $f$ is continuous. I will let you write down a version of this argument for $\mathbb R^{n}$. Does this convince you that linearity plays an important role in the result you have quoted for operators between Banach spaces. Two main ingredients for that result are linearity of the operators and completeness of the space. 

