# Proper handling of witnessing constants in epsilon-delta proofs

1. Sep 21, 2010

### farleyknight

Suppose you had some arbitrary function $f : R^n \to R^p$ and $x \in R^n$. You want to know if it's continuous, so you do some epsilon-delta to find out for sure. However, only the most simple functions permit this without some extra restrictions.

Consider $f(x) = x^2$. To show that $|x - a| < \delta \rightarrow |f(x) - f(a)| < \epsilon$, you'd have to break up $|x^2 - a^2|$ into $|x + a||x - a|$. The next step is something I'm somewhat concerned about, but not even for this particular function. The technique I want to illustrate ought to work for higher powers. If it's a correct technique, it ought to work for the general case.

In general, proofs of continuity for anything but linear combinations would require that you make an assumption $|x - a| < 1$ and then $|x| < 1 + |a|$, by the triangle inequality. In this case, we took the witnessing constant 1 to create the coefficient $(|1 + |a| + a|)$ to be sure that this neighborhood $(|1 + |a| + a|)|x - a| < \epsilon$ holds for the delta we wish to find.

Then when you get around to finding epsilon, you take delta as the min of whatever epsilon and 1, or $\delta = \min( 1, F(\epsilon) )$ where $F(\epsilon)$ is a function of your constant a and any witnessing constants.

My question is: Will this always work? Can I just introduce witnessing constants to replace x whenever I don't want it? (Provided I maintain the inequality.) Will you ever run into trouble with this as a general strategy? I've seen this in all the books I've come across, so I feel the answer is yes, but having someone more experienced will help solidify that.

Surely I will need more advanced techniques for higher level courses, but is there anything I should watch for at my current level (intro analysis course)?

Last edited: Sep 21, 2010