Let E С Шт. We say that a function f : E -» W is continuous at some point x 6 E if, given e > 0, we can find a S(e, x) > 0 such that, whenever у 6 E and ||x — y|| < S(e, x), we have
||f(x)-f(y)||<e.
If f is continuous at every point x 6 E, we say that f is a continuous function on E.
This may be the place to make a comment on vector notation. It is conventional in elementary analysis to distinguish elements of Ж"1 from those in Ш. by writing points of Ш"1 in boldface when printing and underlining them when handwriting. Eventually this convention becomes tedious and, in practice, mathematicians only use boldface when they wish to emphasise that vectors are involved.
Exercise 4.29. After looking at Lemma 1.13 and parts (Hi) to (v) of Lemma 4-9, state the corresponding results for continuous functions. (Thus part (v) of Lemma 4-9 corresponds to the statement that, if A : E —»• M and f : E —»• IRP are continuous at x 6 E, then so is \f.) Prove your statements directly from Definition 4-28.
Suppose that E С ]Rm and f : E —>M. is continuous at x. Show that, if /(t) ф 0 for all t 6 E, then 1// is continuous at x.
Once again we have the following useful observation.