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Sahithyan's S1
Sahithyan's S1 — Mathematics

Continuity

f(z)f(z) is continuous at z0z_0 iff:

limzz0f(z)=f(z0)\lim_{z\to z_0} f(z) = f(z_0)     ϵ>0  δ>0  x  (zz0<δ    f(z)f(z0)<ϵ)\iff \forall{\epsilon>0}\; \exists{\delta>0}\; \forall{x}\; \big(\,|z-z_0|<\delta\implies{\lvert f(z)-f(z_0)\rvert<\epsilon}\,\big)

Conditions

For ff to be continuous on z0z_0, all these conditions are required.

  • ff is defined at z0z_0
  • limzz0f(z)\lim_{z\to z_0} f(z) exists
  • limzz0f(z)=f(z0)\lim_{z\to z_0} f(z) = f(z_0)

Properties

If f,gf,g are continuous at z0z_0, these functions are also continuous at z0z_0:

  • Re(f)\text{Re}(f)
  • Im(f)\text{Im}(f)
  • f\lvert f \rvert
  • f±gf\pm g
  • fgfg
  • fg\frac{f}{g} where g0g\neq 0
  • f(g(z))f(g(z))