For $j \geqslant 2$ an integer, the function $\psi _ { j } : \mathbb { R } \rightarrow \mathbb { R }$ is 1-periodic and defined on $] - 1 / 2,1 / 2 ]$ by $$\forall t \in ] - 1 / 2,1 / 2 ] , \quad \psi _ { j } ( t ) = \max ( 0,1 - j | t | ) .$$ For integers $0 \leqslant k < j$, the functions $\psi _ { j , k } : \mathbb { R } \rightarrow \mathbb { R }$ are defined by $$\forall t \in \mathbb { R } , \quad \psi _ { j , k } ( t ) = \psi _ { j } \left( t - \frac { k } { j } \right) .$$ Show that $\psi _ { j , k } \in \mathscr { C } _ { \text {per} } ( \mathbb { R } )$.
For $j \geqslant 2$ an integer, the function $\psi _ { j } : \mathbb { R } \rightarrow \mathbb { R }$ is 1-periodic and defined on $] - 1 / 2,1 / 2 ]$ by
$$\forall t \in ] - 1 / 2,1 / 2 ] , \quad \psi _ { j } ( t ) = \max ( 0,1 - j | t | ) .$$
For integers $0 \leqslant k < j$, the functions $\psi _ { j , k } : \mathbb { R } \rightarrow \mathbb { R }$ are defined by
$$\forall t \in \mathbb { R } , \quad \psi _ { j , k } ( t ) = \psi _ { j } \left( t - \frac { k } { j } \right) .$$
Show that $\psi _ { j , k } \in \mathscr { C } _ { \text {per} } ( \mathbb { R } )$.