Continuity of function on complex plane

Question

I'm currently taking my first course in Complex analysis and both the lectures and textbook have very minimal examples and it has been extremely frustrating.

Prove that f is continuous on $\mathbb{C}$ when: $$f(z) = \bar{z}$$

Would I be able to use the Cauchy_Riemann Theorem to prove that this is holomorphic and in turn say that it is continuous?

Let $u(x,y) = x$ and $v(x,y) = -iy$.

Therefore, $u_x = 1 , v_y = -i$. Showing that $u_x \neq v_y$.

This shows that the function is not continuous.

Thank you for any guidance.

P.S If anyone has any recommendations on books on undergraduate Complex analysis with a lot of examples please let me know.

Answer 1

A complex-valued function is continuous if and only if both, its real part and its imaginary part are continuous.

$f$ sends $$x+iy\mapsto x-iy$$

$f(x+iy)=u(x,y)+iv(x,y)$, where $u,v$ are real valued functions.

$u(x,y)=x$ is continuous and $v(x,y)=-y$ is also continuous so $f$ is continuous.

Answer 2

We have $|f(z)-f(z_0)|=|\overline{z-z_0}|=|z-z_0| \to 0$ when $z\to 0$. ($|\overline{Z}|=|Z|$).