Method for proving continuity for a complex function
Solution 1:
Using the elementary definition for continuity $|f(z)-f(z_0)|<\epsilon$ for $|z-z_0|<\delta$
so for $|z-z_0|=|\overline{(z-z_0)}|=|\bar z-\bar z_0|=|f(z)-f(z_0)|<\delta$ hence $\epsilon=\delta$. So $f(z)=\bar z$ is continuous.
Solution 2:
$\lim_{z\longrightarrow z_0}\bar{z}=\lim_{z\longrightarrow z_0}x-iy=\lim_{(x,y)\longrightarrow (x_0,y_0)}x-iy=x_0-iy_0$. Since $|x_0-iy_0|\geq0$. then f(z) is continuous.