Approximation of the Heaviside Function whose derivative has a compact support

Question

I am looking for a smooth approximation $H_\delta$ of the Heaviside function, which has the property that $$ \lim_{\delta\rightarrow 0^+}H_\delta =H $$ in the distribution sense, and $$ H_\delta(x)=1,\;\;{\mbox{for}}\;\;x>\delta,\;\;H_\delta(x)=0,\;\;{\mbox{for}}\;\;x<\delta. $$ Now, take the following function, which is a smooth approximation of the Dirac delta function with compact support: $$ f=\left.\exp\left(-\frac{\delta^2}{\delta^2-x^2}\right)\right.,\;\;{\mbox{for}}\;\;|x|<\delta,\;\; {\mbox{and zero elsewhere.}} $$ The integral of $f$ from $-\delta$ to x divided by the integral of $f$ on $\mathbb{R}$ will do the trick. My question is this: is there a simpler way to answer this question than by writing it as an unevaluated integral?

Answer 1

I just responded to my own questions by looking at Robert Israel's answer. The function is

$$ H_\delta=\dfrac{1}{1+{e^{{\tfrac {4 x\delta}{ x^2-\delta^2 }}}}},\;\;{\mbox{for}}\;\;|x|<\delta. $$