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3.2 Solution of Initial Value Problems 81

3.2 Solution of Initial Value Problems

To apply the Laplace transform to the solution of an initial value problem, we must
be able to transform a derivative. This involves the concept of a piecewise continuous
function.

Suppose f (t) is defined at least on [a,b]. Then f is piecewise continuous on [a,b] if:
1. f is continuous at all but perhaps finitely many points of [a,b].
2. If f is not continuous at t 0 in (a,b), then f (t) has finite limits from both sides at t 0 .
3. f (t) has finite limits as t approaches a and as t approaches b from within the
interval.

This means that f can have at most finitely many discontinuities on the interval, and these
are all jump discontinuities. The function graphed in Figure 3.7 has jump discontinuities at t 0 and
t =t 1 . The magnitude of a jump discontinuity is the width of the gap in the graph there. In Figure
3.7, the magnitude of the jump at t 1 is

| lim f (t) − lim f (t)|.
t→t 1 + t→t 1 −

By contrast, let

1/t for 0 < t ≤ 1
g(t) =
0 for t = 0.

Then g is continuous on (0,1], but is not piecewise continuous on [0,1], because
lim t→0+ g(t) =∞.

t
t 0 t 1

FIGURE 3.7 Typical jump discontinuities.

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