Continuity and uniform continuity of $f(z)=\frac{1}{z}$?

Real Analysis experts ,

I have learnt that $f(z)=\frac{1}{z}$ is not uniformly continuous on $0<z\le1$.

My question is:

Is $f(z)=\frac{1}{z}$ uniformly continuous on $0<z<1?$

Thanks for your guidance.