1. ## 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

2. Fix $(h_1,h_2)\in\mathbb{R}^2$.

Since $f(0,0)=g_h(0)=0$, you need to show that when $t$ is small enogh $(th_1,th_2)\in A^c$. If
$h_1$ is 0, it is clear that $(th_1,th_2)\$ is never in $A$. If $h_1\neq 0$, you have only to care of the limit direction in $t$ (+ or -) which makes
$th_1$ positive. The other direction gives the constant function 0. In this case you have to show

$
th_1\geq t^2h_2^2
$

for $t$ small enogh. If $h_1$ is positive it is almost immediate you can do it (when $t\to 0^+$). If $h_1$ is negative and $t\to 0^-$ the inequality

$
h_1\leq th_2^2
$

is also fulfilled for $t$ small enough.