Derivative of piecewise function

Question

Hello more experience people,

I'm working with piecewise functions in Java. There are some points that does not have derivative. Do you have any suggestion for treating at these points?

public StepFunction(double a){
    this.a = a;
}
public double getValueAt(double x){
    if(x < a){
        return 0.;
    }else return 1.;
}

public double getDerivativeAt(double x)
     // implement goes here
     // Is there any suggestion for return value when x = a?
}

Thank you very much!

Answer 1

What you mean is the Heavyside function that is defined as:

O(x) = { 1 for x>0 and 1/2 for x=0 and 0 for x<0 }

And in the general case:

O(x-a) = { 1 for x>a and 1/2 for x=a and 0 for x

The derivative of the Heavyside function is:

d/dx(O(x-a)) = d(x-a)

Hence the derivate in x=a is:

d(a-a) = d(0) = infinity

Therefore one could argue that yous should return

double inf = Double.POSITIVE_INFINITY;

However I suppose it depends on application context. usaly dirach is involved in an integral.