How to extend continous function from $S^1\to S^1$ to $D^2\to D^2$ continously

It might be easier to use polar coordinates. We think of $f$ as a continuous periodic function $f(\theta)$ defined on $\mathbb{R}$ (or $[0,2\pi]$). Then define a function on $D^2$ by $$\overline{f}(r,\theta)=rf(\theta)$$ where $0<r\leq 1$ and $0\leq\theta \leq 2\pi$. Also define it at the origin by $\overline{f}(0)=0$.