Let fj : R n → R, j = 1, .., n be defined as fj (x1, .., xn) = xj . Prove that fj is uniformly continuous in R n.

Question

|fj(x)- fj (y)| = |xj-yj|, which depends on the value of x and y, and so cannot be uniformly continuous. Moreover, I'm wondered if this is true, then is that meeans all mapping function from higher dimensional space to lower dimensional one is uniformly continuous

Answer 1

The mapping $f_j$ is linear hence Lipschitz continuous.

$\|f_j (x)-f_j(y)\| \le \|f_j\| \|x-y\|$.

Answer 2

$|f_j(x)-f_j(y)|=|x_j-y_j| \le ||x-y||$.