# Sequence of continuous functions on [0,1] converging to unbounded function.

I'm working on problem 3 from Exercises 2E of Axler's "Measure, Integration and Real Analysis" which says:

I thought of something like $$f_n (x) = \frac{1}{x + \frac{1}{n}}$$ , but then this converges to $$f(x) = 1/x$$, which isn't defined at $$x = 0$$ so it is not a function $$f: [0,1] \rightarrow \mathbf{R}$$. When I attempt to fix this I lose either the continuity of $$f_n$$ or the convergence to $$f$$.

I'd appreciate any insight

$$f_n(x) = \begin{cases} n^2 x &0 \leq x \leq \frac 1n \\ \frac 1x &\frac 1n \lt x \leq 1. \end{cases}$$
Then $$f_n$$ converges pointwise to $$\frac 1x$$ for $$x \in (0, 1]$$ and $$\forall n~f_n(0)=0.$$