Heaviside forcing in a first-order integrating-factor equation ($y(x)=\mu (x{)}^{-1}[\mu (0)y(0)+{\int }_0^x\mu (t)H(t)dt]$)

For $y^{\prime }+a(x)y=H(x)$ with continuous $a$, the integrating factor $\mu =\exp (\int a)$ gives the displayed solution. It is continuous at the step and solves the differential equation classically on each side; the derivative jumps by one. The value assigned to the Heaviside step function at the single point zero does not affect the integral. A discontinuity in $y$ would instead produce an unintended delta term in its distributional derivative.