absolute value continuity

It is very easy to prove that abs(x) is continuous everywhere using epsilon-delta techniques. Yet, let U = (0,1). This is open in [0, infinity). Yet, f inverse (U) can be said to equal (0,1) U {-1/2} right? This set, call it V is certainly not open in (- infinity, infinity).

Thanks!

Re: absolute value continuity

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**grandunification** It is very easy to prove that abs(x) is continuous everywhere using epsilon-delta techniques. Yet, let U = (0,1). This is open in [0, infinity). Yet, f inverse (U) can be said to equal (0,1) U {-1/2} right? This set, call it V is certainly not open in (- infinity, infinity).

What?

$\displaystyle f^{-1}(U)=(-1,0)\cup(0,1)$ which is open.

Re: absolute value continuity

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Originally Posted by

**Plato** What?

$\displaystyle f^{-1}(U)=(-1,0)\cup(0,1)$ which is open.

The preimages aren't unique. One preimage would be the union of (0,1) with the singleton set {-1/2}. f((0,1)) = (0,1) and f({-1/2}) = 1/2 which is an element of (0/1).

Re: absolute value continuity

Disregard that comment above. I was not really using the proper definition of a preimage. I have to consider ALL elements which map onto the image. Thank-you.