My professor has the idea of a subset being pathwise connected in his notes but he never talked about it. I would appreciate any help.

He says...

Let $\displaystyle D$ be a subset of complex numbers. $\displaystyle D$ is pathwise connected if given any two points $\displaystyle a,b\in D$, there exists a continuous function $\displaystyle \varphi :[0,1]\rightarrow D$, such that $\displaystyle \varphi (0)=a$ and $\displaystyle \varphi (1)=b$.

He then asks us to prove that a rectangle is piecewise connected.