Laplace transform and point of discontinuity

Question

We can apply Laplace transform on, say, a piecewise continuous function $f(t)$. Suppose $f(t)$ is discontinuous at point r. My book says that the value at point $r$, $f(r)$, is not important to the Laplace transform; but the left-handed limit and right-handed limit of $r$ are important to the Laplace transform. Why is this? Why is the function value of any point not important to the Laplace transform?