= Solution
The <Laplace transform> is
$$
\overline y(p)=\int_0^\infty e^{-pt}y(t)\,dt.
$$
A sufficient condition is that $y$ be piecewise continuous on each bounded interval and of exponential order $c$, meaning $|y(t)|\leq Ce^{ct}$ for all sufficiently large $t$. Then the integral is absolutely convergent for $\operatorname{Re}p>c$, by comparison with $e^{-(\operatorname{Re}p-c)t}$.
For the constant function, direct integration gives
$$
\boxed{\mathcal L[1](p)=\frac1p\quad(\operatorname{Re}p>0).}
$$
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