Let $a$ and $b$ be two real numbers. Show that the function $\left\lvert\, \begin{array} { r l l } \mathbb { R } _ { + } ^ { * } & \rightarrow & \mathbb { R } \\ t & \mapsto & a \mathrm { e } ^ { t } + b \end{array} \right.$ belongs to $E$ if and only if $a = b = 0$, 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.
Let $a$ and $b$ be two real numbers. Show that the function $\left\lvert\, \begin{array} { r l l } \mathbb { R } _ { + } ^ { * } & \rightarrow & \mathbb { R } \\ t & \mapsto & a \mathrm { e } ^ { t } + b \end{array} \right.$ belongs to $E$ if and only if $a = b = 0$, 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.