# Laplace Transform of Derivative/Discontinuity at t = 0

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## Theorem

Let $f: \R \to \R$ or $\R \to \C$ be a continuous function, differentiable on any interval of the form $0 < t \le A$.

Let $f$ be of exponential order $a$.

Let $f'$ be piecewise continuous with one-sided limits on said intervals.

Let $\laptrans f$ denote the Laplace transform of $f$.

Let $f$ fail to be continuous at $t = 0$, but let:

- $\displaystyle \lim_{t \mathop \to 0} \map f t = \map f {0^+}$

exist.

Then $\laptrans f$ exists for $\map \Re s > a$, and:

- $\laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f {0^+}$

## Proof

## Sources

- 1965: Murray R. Spiegel:
*Theory and Problems of Laplace Transforms*... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $5$. Laplace transform of derivatives: Theorem $1 \text{-} 7$