Lecture Notes in Differential Equations - Bruce E. Shapiro

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Figure 32.1: A piecewise continuous function. This function is piecewise
continuous in three intervals [−1, 1/2], [1/2, 3/2], [3/2, 2] and the function
approaches a finite limit at the endpoint of each interval as the endpoint is
approached from within that interval.
2
3
2
1
1
2
1
1
2
1
2
1
2
1
3
2
2
1
Theorem 32.5 (Integrability). If f(t) is piecewise continuous on (a, b)
and f(t) approaches a finite limit at the endpoint of each interval as it is
approached from within the interval then ∫ b
f(t)dt exists, i.e., the function
a
is integrable on (a,b).
Theorem 32.6 (Existence of Laplace Transform). Let f(t) be defined for
all t ≥ 0 and suppose that f(t) satisfies the following conditions:
1. For any positive constant A, f(t) is piecewise continuous on [0, A].
2. f(t) is Exponentially Bounded.
Then the Laplace Transform of f(t) exists for all s > a.
Proof. Under the stated conditions, f is piecewise continuous, hence integrable;
and |f(t)| ≤ Ke at for some K, a, M for all t ≥ M. Thus
L[F (t)] =
=
∫ ∞
0
∫ M
0
e −st f(t)dt (32.12)
e −st f(t)dt +
∫ ∞
M
e −st f(t)dt (32.13)
Since f is piecewise continuous, so is e −st f(t), and since the definite integral
of any piecewise continuous function over a finite domain exists (it is the
area under the curves) the first integral exists.