Let f ( x ) = { z 3 − 1 z 2 + z + 1 i f | z | ≠ 1 − 1 + i 3 2 i f | z | = 1 is f continous in z 0 = 1 + 3 i 2 I think to f is not continous at z 0 , i try using the sequence criterion for continuity searching a sequence { z n } such that lim n → ∞ z n = z 0 but lim n → ∞ f ( z n ) ≠ f ( z 0 ). but i cant find that sequence, ill be very grateful for any hint or help to solve my problem.

Question

Let  f  (  x  )  =      {                                                                      z                3                            −              1                                                      z                2                            +              z              +              1                                                i          f                                      |                    z                      |                    ≠          1                                                                                    −              1              +              i                              3                                      2                                    i          f                                      |                    z                      |                    =          1                        is   f continous in       z    0    =            1      +              3            i        2  I think to   f is not continous at       z    0  , i try using the sequence criterion for continuity searching a sequence   {      z    n    } such that       lim          n      →      ∞            z    n    =      z    0   but       lim          n      →      ∞        f  (      z    n    )  ≠  f  (      z    0    ). but i cant find that sequence, ill be very grateful for any hint or help to solve my problem.

belidla5a · Accepted Answer

Note that, if       |    z      |    ≠  1,  f  (  z  )  =                    z        3            −      1                      z        2            +      z      +      1        =            (      z      −      1      )      (              z        2            +      z      +      1      )                      z        2            +      z      +      1        =  z  −  1and that      lim          z      →              z        0              z  −  1  =            −      1      +              3            i        2    .So,   f is continuous at       z    0  .Actually, this argument only works for       z    0  . In fact,   f is discontinuous at every other point of the unit circle.

Let f(x)={((z^3−1)/(z^2+z+1) if |z|!=1 (−1+i sqrt(3))/2 if |z|=1) is f continous in z_0=(1+sqrt(3)i)/2

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