Differentiable Function as Distribution

Theorem

Let $T \in \map {\DD'} \R$ be a distribution.

Let $f : \R \to \R$ be a continuously differentiable real function.

Suppose $T_f$ is a distribution identified with $f$.

Then $T_f' = T_{f'}$.

Proof

Let $\phi \in \map \DD \R$ be a test function with a support on $\closedint a b$.

Then:

\(\ds \map {T'_f} \phi\)

\(=\)

\(\ds -\map {T_f} {\phi'}\)

Definition of Distributional Derivative

\(\ds \)

\(=\)

\(\ds -\int_{-\infty}^\infty \map f x \map {\phi'} x \rd x\)

\(\ds \)

\(=\)

\(\ds -\int_a^b \map f x \map {\phi'} x \rd x\)

Definition of Test Function, Integral of Compactly Supported Function

\(\ds \)

\(=\)

\(\ds -\paren {\map f b \map \phi b - \map f a \map \phi a} + \int_a^b \map \phi x \map {f'} x \rd x\)

Integration by Parts

\(\ds \)

\(=\)

\(\ds \int_a^b \map \phi x \map {f'} x \rd x\)

Definition of Test Function

\(\ds \)

\(=\)

\(\ds \map {T_{f'} } \phi\)

$\blacksquare$

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.2$: A glimpse of distribution theory. Derivatives in the distributional sense