Thread: characteristic function

A = {(x,y) in R^2: x > 0 and 0< y < x^2}

Define f:R^2 -->R by f(x,y) = 0 if (x,y) not in A
= 1 if (x,y) in A

Fix h in R^2. Define g: R -->R with g_h(t) = f(t.h)
Show that g is continuous at 0
I dont understand how the funtion g_h(t) = f(t.h) work?

I just use the defintnition of continuity,
This is what I done so far:

given e > 0, there exists d > 0 such that |f(t.h) - 0|< e
But I dont know how to get from |f(t.h) - 0| to |f(t.h) - 0|< e

Fix .

Since , you need to show that when is small enogh . If
is 0, it is clear that is never in . If , you have only to care of the limit direction in (+ or -) which makes
positive. The other direction gives the constant function 0. In this case you have to show



for small enogh. If is positive it is almost immediate you can do it (when ). If is negative and the inequality



is also fulfilled for small enough.