For use when asking questions about piecewise continuous functions, their properties, or the functional branch itself.
Piecewise continuous functions are functions that have the following properties:
The left and right hand limits exist everywhere on the function.
The function is discontinuous for at least one point.
A piecewise continuous function can always be expressed as two functions f + g where f is continuous and g is a piecewise constant constant function.
g is sometimes referred to as the jump series or jump component of the piecewise continuous function. g can only be determined up to some constant c. The formula for finding g is:
$$g(x) = \sum_{n=0}^{JC(x)} (\lim_{a -> JL(n)^+} f(a) - \lim_{a -> JL(n)^-} f(a))$$
Where, JC(x) is the function counting the number of jump discontinuities between $0$ and x, and JL(x) is the discrete function returning the x coordinate of the X'th discontinuity. JC returns a negative count for values of negative x.
Another important operation involving piece-wise continuous functions are two alternative calculus operators, implied by the following:
The indefinite integral "area function" always exists for piece-wise continuous functions, yet the anti-derivative doesn't always exist.
There is no anti-indefinite integral such that the indefinite integral returns the original function, or some analogue fully defined for all $x \in R$
The result is two operators: the implied integral and the implied derivative which fulfill 1 and 2. The implied derivative is nothing more than the one sided derivative limit which is hardly useful or noteworthy. The implied integral on the other hand, has a much deeper change and is defined as a form similar to that of symbolic integration, but where any "symbol" of the form "$\lfloor f(x) \rfloor$" is held fixed and held fixed in the same manner that y is held fixed when integrating the multivariate function f(x,y) with respect to x. The implied integral will vary by piece wise constant functions of x, rather than the usual constants normal integration varies by.
The first fundamental theorem of implied calculus states:
Any integral of f is equal to the implied integral of f minus the jump series of the implied integral of f.
In this way, many integrals can be changed into a problem of finding the jump series.
