Proving a mapping from a higher dimension to a lower dimension is continuous? How does one show a mapping is continuous from a higher dimension to a lower dimension. For example:
Let f : $\mathbb{R}^2$ $\rightarrow$ $\mathbb{R}$, (x,y) $\rightarrow$ x. How would you show this is continuous.
My attempt was to use the definition of continuous:
That is: f is continuous at $x_0 \in$ its domain if and only if for every $\epsilon$ > 0, there exists a $\delta$ > 0 such that for all x $\in$ its domain A, d(x,$x_0$) < $\delta$ implies $\gamma$(f(x),f($x_0$)) < 0.
So set y = $y_0$ and then the proof should be vascuous. However, I feel that there is something wrong with this approach.

You have $$0 \le \vert x \vert \le \sqrt{x^2 +y^2}.$$ Now take $\epsilon > 0$, for $x^2 + y^2 \le \epsilon^2$ you have $0 \le \vert x \vert \le \epsilon$ allowing you to conclude that $\delta = \epsilon^2$ is convenient for the $\epsilon$-$\delta$ proof. 
