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largest interval on which f(z) continuous?

Hi,

just wanted to check:

Qn: find the largest region in the complex plain on which

f(z)=Re(z)Im(z)

is continous?

i get the entire complex plane excluding z=0.

(since

f(z)=w=xy

) (akin to this in catesian co-ordinates is the hyperbolic parabola)

book sais entire complex plane?

(edited 10 years ago)

Reply 1

msmith2512

If the function isn't continuous at z=0 then show me a path from a general point z=x + iy to z=0 which isn't continuous.

Reply 2

Hasufel

you can approach (0,0) from 4 different directions, all of which end up having a limit of 0 (+/- xreal/imaginary axis.)

Reply 3

msmith2512

Original post by Hasufel

you can approach (0,0) from 4 different directions, all of which end up having a limit of 0 (+/- xreal/imaginary axis.)

You can approach z=0 in infinitely many different directions and ways - not just 4. You ave to show that it is continuous for every single direction and way.

Try f(z) = xy/(x^2+y^2) - this isn't continuous as the results differ as you approach (0,0) along the lines x=0 and x=y.