MATH.17:5.4 - Change a function while preserving its increments

Given a function f from the real numbers to the real numbers and a chosen point c, construct a function g that fixes c and preserves every increment of f. The requirements are g(c)=c and g(x)-g(y)=f(x)-f(y) for every x,y.

Set y=c in the second requirement. It forces g(x)=f(x)-f(c)+c. This defines a real-valued function; substituting c establishes the fixed point, and subtracting its values at x and y cancels the added constant and preserves the required increment. Thus the formula supplies the unique function under these requirements.

The transformation takes f itself as an input and returns g. Fixing a point and preserving increments do not settle an additional question about composition or cost. The repetition case in :5.2 shows how different requested changes lead to different operations on the same input rules.