$f(z) = 2z + 3\overline z$, show using the limit definition that $f$ is continuous at every point $a\in\mathbb C$.

Question

For the function $f : \mathbb C\to\mathbb C$ defined by $f(z) = 2z + 3\overline z,\forall z\in\mathbb C$, show using the limit definition, by finding an inequality between $|f(z) − f(a)|$ and $|z − a|$, that $f$ is continuous at every point $a\in\mathbb C$.

When I expand $|f(z) - f(a)|$ I get $2(z-a) + 3(\overline z - \overline a)$. I am unsure how to complete the proof after this?

Answer 1

Hint: $$|2(z-a)+3(\bar{z}-\bar{a})|\leq|2(z-a)|+3|\bar{z}-\bar{a}|\leq2|z-a|+3|z-a|$$