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Corollary 7.5.1 The solution has all derivatives of all orders for t > 0,

even if φ is not differentiable. We can say therefore that all solutions
become smooth as soon as diffusion takes effect. There are no singu-
larities, in sharp contrast to the wave equation.

Piecewise Continuous Initial Data. Notice that the continuity of φ(x) was used
in only one part of the proof. With an appropriate change we can allow that φ(x) to have
a jump discontinuity. [Consider, for instance, the initial data for Q(x, t). A function φ(x)

is said to have a jump at x 0 if both the limit of φ(x) as x → x 0 from the right exists
−
+
[denoted φ(x )] and the limit from the left [denoted φ(x )] exist but these two limits are
0 0
not equal. A function is called piecewise continuous if in each finite interval it has only a
finite number of jumps and it is continuous at all other points.

Theorem 7.5.2 Let φ(x) be a bounded function that is piecewise con-
tinuous. Then (7.34) is an infinitely differentiable solution for t > 0
and
1
+
−
lim u(x, t) = (φ(x ) + φ(x ))
t→+0 2
for all x. At every point of continuity this limit equals φ(x).

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