I found this text online:
"In general, integrating the $\delta$ function or one of its integrals makes it smoother. Differentiating it increases the discontinuities. For example $\int\delta $ is discontinuous itself. $\int \int \delta $ is continuous but with a discontinuous first derivative. $\int \int \int \delta$ is continuous, but with a discontinuous second derivative, etc..."
I agree that $\int\delta $ is certainly discontinuous since it equates to the Heaviside function. However I do not agree that $\int \int \delta $ is continuous since wouldn't this simply be the Ramp function ($\int H(x)$) And isn't the Ramp function also discontinuous at $x = 0$?