I'm looking for where I've made a mistake, since I'm reaching a contradiction after applying key renewal theorem and differentiating through the convolution.
Let $(X_t)$ be a renewal process with continuous inter-arrival times $Z_1,Z_2,\dots$ with pdf $f$, cdf $F$ and finite mean $\mu$. Let $\overline F(t) := \mathbb{P}(Z_1 > t) = 1-F(t)$, and we have $\mu = \int_0^\infty \overline F(t) \mathrm{d}t$. Finally write $m(t) = \mathbb{E}[X_t]$ be the renewal function.
Then $\overline F$ is integrable and non-increasing so by key renewal theorem as $t \to \infty$ $$ (\overline F \star m' )(t) := \int_0^t \overline F(t-u)m'(u) \mathrm{d}u \ \to \dfrac{1}{\mu} \int_0^\infty \overline F(t) \mathrm{d}t = 1 $$
But by Leibniz rule, (/using this Derivative of convolution), and then by renewal equation ($m = F + m \star f$) we have $$ \overline F \star m' = (\overline F \star m)' = \overline F' \star m = -f \star m = F-m $$
so again taking $t \to \infty$, $F(t) \to 1$ since $F$ is a cdf, and so $$ m(t) = F(t) - (\overline F \star m' )(t) \to 1 - 1 = 0 $$
which is clearly a contradiction, since $X_t$ is a counting process and not necessarily $0$ throughout.
