My professor has this exercise in his notes. I may be able to prove it but I don't know what he pseudo-sine function is.

He states:

Let $\displaystyle \sigma$ be the pseudo-sine function. Let $\displaystyle f(x)=x\sigma (1/x)$ for $\displaystyle x\not= 0$. Prove $\displaystyle f$ has a removable discontinuity at $\displaystyle 0$.