Well known and conventional defintion of Heaviside function is
H(x) = 0, x < 0 H(x) = 1/2, x = 0 H(x) = 1, x > 0 Mathematica uses instead unconventional "unit step" for its $HeavisideTheta[x] $ function
S(x) = 0, x < 0 S(x) = 1, x > 0 How to use in Mathematica proper Heaviside function with normal definiton $H(x)$?
myHeaviside[x_] := Which[x < 0, 0, x == 0, 1/2, x > 0, 1] Note that the derivative, computed as a limit, is properly a representation of a DiracDelta function (though its integral and higher derivatives might not be appropriately represented):
Limit[(myHeaviside[x + ε] - myHeaviside[x])/ε, ε -> 0] $\begin{cases} \infty & x=0 \\ 0 & \text{True} \end{cases}$
DiracDelta, I believe. $\endgroup$ As the link in the question states, you can get the desired definition as follows:
heaviside[x_] := 1/2 (1 + Sign[x]) Here is a test:
heaviside /@ {-1, 0, 1}
{0, 1/2, 1}
UnitStepis a separate function, and the link you provided states thatHeavisideThetacan be defined in different ways. $\endgroup$