Let $(\phi_i)_{1\leq i\leq m}$ be functions from $\mathbb{R}$ to itself, and set
\begin{equation} f= \begin{cases} \phi_1 &\text{ if *case 1*}\\ \phi_2 &\text{ if *case 2*}\\ \vdots & \vdots \\ \phi_m &\text{ if *case m*}. \end{cases} \end{equation}
Is it correct to say, "$f$ is a piecewise function with $m$ pieces" Or is there a different term to refer to each line above? By "piece" I mean both the pair of $\phi_i$ and * case i *
Also: Is there a name to refer to the individual functions $(\phi_i)$? e.g. would it be proper to say "$f$ is a piecewise function, where each piece is continuous?"