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