Dual Space of a Vector Space Question

Hello everyone I have a question about Dual Spaces. Let $\displaystyle X$ be an infinite dimensional vector space over $\displaystyle \mathbb{C}$, and let

$\displaystyle \{ x_1, \ldots, x_n \} $ be a subset of $\displaystyle X$, which is linearly independent. Then we can define the unique linear functionals $\displaystyle $f_i : X \rightarrow \mathbb{C}$ which give $\displaystyle f_i (x_j) = 1$ if $\displaystyle i=j,$ 0 otherwise.

My question is can we choose the $\displaystyle f_i$'s to be continuous? I was asked this question and I am not sure what the answer is. I take it if we have an infinite basis $\displaystyle \{ x_i\}_{i \in I}$ of X then we cannot have every $\displaystyle f_i$ continuous because then surely every linear functional would then be determined by those continuous functions, i.e. it is a linear combination of finitely many of them, hence is continuous. And there exist unbounded linear functionals (?). Any help with this would be appreciated! Thanks