Type of discontinuity

[jwpaine]

for the function: \(\displaystyle \L f(x) = \frac{x\,+\, |x|}{2x}\)

find the domain and classify any points of discontinuity as removable, jump, infinite or oscillatory.

Well the domain is R: x =/ 0 so the function has a vertical asymptote at x = 0
but I'm not sure what type of discontinuity it is... I would like to say jump, because for all x < 0, f(x) = 0 and for all x > 0, f(x) = 1.... but it is undefined at x = 0. It obviously wouldn't be oscillatory or infinite, so I would say either removable (because of x = 0) or jump because of the jump from f(x) = 0 to f(x) = 1

Any hints?
Thanks,
John.

 

[Deleted member 4993]

for the function: \(\displaystyle \L f(x) = \frac{x\,+\, |x|}{2x}\)

find the domain and classify any points of discontinuity as removable, jump, infinite or oscillatory.

Well the domain is R: x =/ 0 so the function has a vertical asymptote at x = 0
but I'm not sure what type of discontinuity it is... I would like to say jump, because for all x < 0, f(x) = 0 and for all x > 0, f(x) = 1.... but it is undefined at x = 0. It obviously wouldn't be oscillatory or infinite, so I would say either removable (because of x = 0) or jump because of the jump from f(x) = 0 to f(x) = 1

Any hints?
Thanks,
John.

It is not removable discontinuity.

Removable discontinuities are of the type (x^-4)/(x-2) where x = 2 is a removable discontinuity (the slope does not change).

 

[galactus]

It certainly has a jump.

\(\displaystyle \L\\\lim_{x\to\0^{-}}\frac{x+|x|}{2x}=0\)

\(\displaystyle \L\\\lim_{x\to\0^{+}}\frac{x+|x|}{2x}=1\)