Implied Integration

The implied integral is an operator alternative to the integral but similar enough to be useful. The basic idea of what it is stems from the following two rules (or as we view, axioms) of regular calculus:

1. The derivative of a constant function is 0.
2. The indefinite integral varies by constant functions.

(We accept that multivariate calculus complicates things heavily)

Implied calculus can in general by thought broadly as all pairs of "derivatives" and "integrals" where these two statements are changed in some way. The simplest change is to change "constant" to mean something else. The implied integral/derivative has the following two rules:

1. The implied derivative of a piece-wise constant function is 0.
2. The implied integral varies by piece-wise constant functions.

The implied integral has a nice property that any known normal integral with various arbitrary constants has the same implied integral but with arbitrary piece-wise constant functions rather than arbitrary constants. Furthermore, the following (albeit unproven) conjecture links implied integration to normal integration.

All continuous implied integrals of a function are indefinite integrals of the function.

If we use implied integration to solve implied differential equations (it's the same as differential equations but with implied integrals and derivatives) than the same property of arbitrary constants follows the same logic as with basic integrals. From this it follows the broader conjecture which all the more useful and intriguing.

All continuous solutions to implied differential equations are solutions to the corresponding differential equation.