Question 12.9: a) Let f be a contin uous function on R \ { a }, which is al......
a) Let f be a contin uous function on R \ { a }, which is also continuously differentiable on the intervals (-∞, a] and [a, +∞). Prove that
D f=T_{f^{\prime}}+[f]_a \delta_a
where D f denotes the distributional derivative of the function f, while T_{f^{\prime}} is the regular distribution defined by the classical derivative f’ of f, see equation (12.2). As usual,
\left\langle T_f, \varphi\right\rangle=\int_O f(x) \varphi(x) d x \quad(\varphi \in \mathcal {D} (O)) (12.2)
[f]_a=f(a+)-f(a-)
is the jump of f at the point a.
b) If f is a piecewise continuously differentiable function on R with isolated discontinuities at the points a_j, j \in J, J a finite or infinite subset of N, then
D f=T_{f^{\prime}}+\underset{j∈J}{\sum}[f]_{a_j} \delta_{a_j}
Remark 12.9.1 In a), by assumption, f’ exists and is continuous on the set R \ {a}, but not in the point a. Since a point is a set of measure zero, the classical derivative f’ of the function f defines a locally integrable function on R.
Clearly it is enough to prove part a). To that end, for \varphi \in \mathcal {D} ( \mathbf{R}) we have using (12.4)
\left\langle D^\alpha T, \varphi\right\rangle=(-1)^{|\alpha|}\left\langle T, \frac{\partial^\alpha \varphi}{\partial x^\alpha}\right\rangle \quad(\varphi \in \mathcal {D} (O)) (12.4)
Since \varphi(a)=\left\langle\delta_a, \varphi(x)\right\rangle, we obtain the statement.