38Lecture 6If∆zis real, then∆z=∆zand the difference quotient is 1.If∆zispurely imaginary, then∆z=−∆zand the quotient is−1.Hence, the limitdoes not exist as∆z→0.Thus,zis not differentiable. In real analysis,construction of functions that are continuous everywhere but differentiablenowhere is hard.The proof of the following results is almost the same as in calculus.Theorem 6.1.Iffandgare differentiable at a pointz0,then(i).(f±g)′(z0) =f′(z0)±g′(z0),(ii).(cf)′(z0) =cf′(z0) (cis a constant),(iii). (fg)′(z0) =f(z0)g′(z0) +f′(z0)g(z0),(iv).fg′(z0) =g(z0)f′(z0)−f(z0)g′(z0)(g(z0))2ifg(z0)̸= 0,and(v).(f◦g)′(z0) =f′(g(z0))g′(z0),providedfis differentiable atg(z0).Theorem 6.2.Iffis differentiable at a pointz0,thenfis continuousatz0.A functionfof a complex variable is said to beanalytic(orholomorphic,orregular) in an open setSif it has a derivative at every point ofS.IfSisnot an open set, then we sayfisanalytic inSiffis analytic in an openset containingS.We callfanalytic at the pointz0iffis analytic in someneighborhood ofz0.It is important to note that while differentiability isdefined at a point, analyticity is defined on an open set. If a functionfisanalytic on the whole complex plane, then it is said to beentire.