Function which is Zero except on Countable Set of Points is Null

Theorem

Let $S \subseteq \R$ be a subset of $\R$ such that $S$ is countable, either finite or countably infinite.

Let $f: \R \to \R$ be a real function such that:

$\forall x \in \R \setminus S: \map f x = 0$

That is, except perhaps for the elements of $S$, the value of $f$ is zero.

Then $f$ is a null function.

Proof

This is an instance of Measurable Function Zero A.E. iff Absolute Value has Zero Integral.

$\blacksquare$

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Special Functions: $\text {IX}$. Null functions