
continuous metric spaces
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 $
x1x2 <$\displaystyle \epsilon $
We can define $\displaystyle \delta $ = the square root of ($\displaystyle \epsilon $^2  (y2y1)^2)
Since there exists a delta for any epsilon we choose, implies that p is continuous.
Is this correct...

Looks quite correct to me.