Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $n \in \mathbb { N }$ and $h \in \mathbb { R } _ { + } ^ { * }$, we set $g _ { n } ( h ) = \operatorname { sinc } \left( \frac { x _ { n } - m } { h } \right) \mathbb { P } \left( X = x _ { n } \right)$.\\
Show that the function $g _ { n }$ extends to a function $\tilde { g } _ { n }$ defined and continuous on $\mathbb { R } ^ { + }$.