Show that

1. the projection map p : R^2 → R given by p(x; y) := x is continuous.

.

My proofs:

We want to show that $\displaystyle \forall$ $\displaystyle \epsilon$ > 0, $\displaystyle \exists $$\displaystyle \delta$>0 such that:

d(x,y)<$\displaystyle \delta$ $\displaystyle \Rightarrow$ d(f(x),f(y)) <$\displaystyle \epsilon$

(x1,y1) (x2,y2) are points in R^2

1) d(x1,x2) < $\displaystyle \epsilon $

|x1-x2| <$\displaystyle \epsilon $

We can define $\displaystyle \delta $ = the square root of ($\displaystyle \epsilon $^2 - (y2-y1)^2)

Since there exists a delta for any epsilon we choose, implies thatis continuous.p

Is this correct...