If $$ x_1(t) = \left\{ \begin{align} 0, & \quad t \ne 0 \\ 1, & \quad t = 0 \end{align} \right. \quad \quad x_2(t) \equiv 0 $$ and $y_k(t) = \int_{-\infty}^t x_k(\tau) \mathrm{d}\tau,~ k = 1, 2$
Is $y_1(t)$ identical with $y_2(t)$? What's the result of $\frac{\mathrm{d}}{\mathrm{d}t} y_1(t)$?
The two are the same, and $\frac{dy_1}{dt}$ is identically zero. This does not contradict the (Riemann) fundamental theorem of calculus because the FTC requires the integrand to be continuous at the point where you differentiate.