Laplace transformation of a piecewise function

Question

How to find Laplace transformation of $$ f(t) = \begin{cases} 1, & t\geq 0, t\neq 1, t\neq 2 \\ 3, & t=1 \\ 4, & t=2 \end{cases} $$

Answer 1

Let $f(x)$ be as stated in the question and let $\hat{f}(x)$ differ from $f(x)$ in finitely many points. Then:

$\int\limits_a^b f(x) dx = \int\limits_a^b \hat{f}(x) dx$ for all $[a,b] \subset \mathbb{R}$.

This should help you. Take $\hat{f}(x) =1$ then we have $f(x) = \hat{f}(x)$ except for $x = 1,2$ which is a finite amount.