Is the functional $f\mapsto\int_0^1 f(x)x^{-1/2}dx$ continuous in $L^2[0,1]$?

Question

I was looking for examples of linear functionals which are continuous in $L^2[a,b]$ but not $L^1[a,b]$. My first thought was to try $F(f)=\int_0^1f(x)x^{-1/2}dx$, but I had trouble showing it was continuous in $L^2[0,1]$. However, it was easy to show that $f\mapsto \int_0^1f(x)x^{-1/3}dx$ was an example by using H$\ddot{\text{o}}$lder's inequality (or more specifically, the Cauchy-Schwarz inequality in this case). After some thought, I concluded that the linear functional $f\mapsto\int_0^1f(x)x^{-1/2+\epsilon}dx$ is continuous and $f\mapsto\int_0^1f(x)x^{-1/2-\epsilon}dx$ is discontinuous in $L^2[0,1]$ for any $\epsilon\in(0,1/2)$. However, I'm still a little stuck on determining whether $F(f)=\int_0^1f(x)x^{-1/2}dx$ is continuous. My instincts tell me it should be continuous, but H$\ddot{\text{o}}$lder's inequality doesn't work as nicely as it did when the exponent on $x$ is greater than $-1/2$.