Show that the function f given by $f(\mathbf{x}) = |\mathbf{x}|$ is continuous on $\mathbb{R}^n$

Question

amistre64 · Answer

hmmm  thats vector notation isnt it

anonymous · Answer

Yes, x would be an n-dimensional vector

amistre64 · Answer

then |x| = sqrt(x1^2+x2^2+...+xn^2)

anonymous · Answer

I had that, from there would it be ok to just take the limit to a, and substitute it in since the square root is continuous?

amistre64 · Answer

if the right and left limits, or in this case the disk, ball, something, limits exist

and the value of f(x) is equal to the limit at x, then it is continuous right?

anonymous · Answer

Yes

amistre64 · Answer

might want to determine if there is any special handling at the "origin" since it might be hard to express that as a onesided limit

amistre64 · Answer

we could go:  d = sqrt(....); and d^2 = (.....) maybe

zarkon · Answer

$$\bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|$$

zarkon · Answer

I would do an $\epsilon,\delta$ proof using the reverse triangle inequality (above)

anonymous · Answer

Alright I'll give that a shot :)

If I run into trouble I'll tell you guys :D