For all $x \in \mathbb { R } _ { + } ^ { * }$ and all $t \in \mathbb { R } _ { + } ^ { * }$, we denote $k _ { x } ( t ) = \mathrm { e } ^ { \min ( x , t ) } - 1$ where $\min ( x , t )$ denotes the smaller of the real numbers $x$ and $t$. Draw a graph of the function $k _ { x }$. Show that $k _ { x }$ belongs to $E$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.
For all $x \in \mathbb { R } _ { + } ^ { * }$ and all $t \in \mathbb { R } _ { + } ^ { * }$, we denote $k _ { x } ( t ) = \mathrm { e } ^ { \min ( x , t ) } - 1$ where $\min ( x , t )$ denotes the smaller of the real numbers $x$ and $t$. Draw a graph of the function $k _ { x }$. Show that $k _ { x }$ belongs to $E$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.