English: The function ${\displaystyle f:\mathbb {R} \to \mathbb {R} ,x\mapsto \min(1,x^{2})}$ is an example of a piecewise monotonic and piecewise differentiable function. The intervals the domain is partioned in differ by property, they are ${\displaystyle (-\infty ,0]}$ and ${\displaystyle [0,+\infty )}$ for monotonicity, and ${\displaystyle (-\infty ,-1]}$, ${\displaystyle [-1,+1]}$, and ${\displaystyle [+1,\infty )}$ for differentiability,

The function definition doesn't involve an explicit case distinction. While the usual definition ${\displaystyle \min(x,y)=\left\{{\begin{array}{ll}x&{\text{ if }}x\leq y\\y&{\text{ if }}y\leq x\\\end{array}}\right.}$ uses a case distinction, so does addition on natural numbers ${\displaystyle x+y=\left\{{\begin{array}{ll}x&{\text{ if }}y=0\\s(x+y')&{\text{ if }}y=s(y')\\\end{array}}\right.}$ which is employed, directly or indirectly, by all arithmetic operations on ${\displaystyle \mathbb {N} }$, ${\displaystyle \mathbb {Z} }$, ${\displaystyle \mathbb {Q} }$, ${\displaystyle \mathbb {R} }$, and ${\displaystyle \mathbb {C} }$. Hence, indirect use of case distinctions cannot be avoided in any function definition whatsoever.

The function ${\displaystyle g:\mathbb {R} \to \mathbb {R} ,x\mapsto x^{2}}$ (not depicted) is piecewise monotonic but built from arithmetic operations only.