The easiest way to understand an auxiliary function is as a temporary helper function that we introduce to make another problem easier to solve or prove.
“Auxiliary” simply means providing additional help or support. Therefore, an auxiliary function is not one special class of mathematical function. Its exact form depends on the problem.
For example, suppose we want to prove a statement about a function f(x)f(x). We might construct
g(x)=f(x)−L(x),g(x)=f(x)-L(x),
where L(x)L(x) is another expression chosen so that g(x)g(x) has useful properties. We can then apply a theorem to gg, obtain information about it, and translate that information back into a statement about ff.
A familiar example occurs in proofs using Rolle's Theorem or the Mean Value Theorem. We sometimes construct an auxiliary function specifically so that it has equal values at two endpoints. Rolle's Theorem can then be applied to that new function.
So rather than asking, “What does an auxiliary function look like?”, think:
Original problem → construct a helpful function → apply a known theorem or technique → use the result to solve the original problem.
It is essentially the mathematical equivalent of introducing a useful intermediate tool. Once you understand why the new function was constructed, the idea of an auxiliary function becomes much more intuitive.
