Optional function arguments

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For that usecase you could make the existing function a wrapper for a new function.
I am using C-syntax here just to demonstrate what I mean:

void existingFunction(int arg1, int arg2)
{
    // Here used to be the old code - remove it and replace it with  this call
    newFunction(arg1, arg2, DEFAULT_VALUE_FOR_ARG3);
}

void newFunction(int arg1, int arg2, int arg3)
{
    // insert the source code of your existing function here and
    // add handling for arg3
}

That way you can keep existing calls and call the new function wherever you need the optional argument.