An unbounded continuous function on $(0,1)$ that is in $L^p(0,1)$ for $1\le p

An unbounded continuous function on $(0,1)$ that is in $L^p(0,1)$ for $1\le p

So this is a question on an old qualifying exam I was going over. Give an
example of a function $g$ such that $g$ is continuous and unbounded on
$(0,1)$ and that $g \in L^p(0,1) $ for $1 \le p < \infty$. I haven't made
any headway with this.
My roommate suggested perhaps doing some construction like enumerating the
rationals in $(0,1)$ and putting a spike on the nth rational of height n
and width $\frac{1}{n2^{n-1}}$. Then the sum of the areas is the geometric
series so it should sum to 1. But there are a lot of details I'm unsure
about, ie that end function would be a countable sum of continuous
function which doesn't seem like it needs to be a continuous function
(fourier series etc). And it isn't apparent that this would be bounded for
all $p < \infty$.
Anywho, I am stumped. Does anybody have any ideas or would like to provide
a hint?