Limits of Real and Imaginary Parts

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Theorem

Let $f: D \to \C$ be a complex function, where $D \subseteq \C$.

Let $z_o \in D$ be a complex number.


Suppose $f$ is continuous at $z_0$.


Then:

$(1): \quad \displaystyle \lim_{z \to z_o} \operatorname{Re} \left({f \left({z}\right) }\right) = \operatorname{Re} \left({ \lim_{z \to z_o} f \left({z}\right) }\right)$
$(2): \quad \displaystyle \lim_{z \to z_o} \operatorname{Im} \left({f \left({z}\right) }\right) = \operatorname{Im} \left({ \lim_{z \to z_o} f \left({z}\right) }\right)$


Here, $\operatorname{Re} \left({f \left({z}\right) }\right) $ denotes the real part of $f \left({z}\right)$, and $\operatorname{Im} \left({f \left({z}\right) }\right) $ denotes the imaginary part of $f \left({z}\right)$.


Proof

By definition of continuity:

$\forall \epsilon > 0: \exists \delta > 0: \left\vert{z - z_0}\right\vert < \delta \implies \left\vert{f \left({z}\right) - f \left({z_0}\right)}\right\vert < \epsilon$

Given $\epsilon > 0$, we find $\delta > 0$ so for all $z \in \C$ with $\left\vert{z - z_0}\right\vert < \delta$:

\(\displaystyle \epsilon\) \(>\) \(\displaystyle \left\vert{f \left({z}\right) - f \left({z_0}\right) }\right\vert\)
\(\displaystyle \) \(\ge\) \(\displaystyle \left\vert{ \operatorname{Re} \left({f \left({z}\right) - f \left({z_0}\right) }\right) }\right\vert\) by Modulus Larger than Real Part
\(\displaystyle \) \(=\) \(\displaystyle \left\vert{ \operatorname{Re} \left({f \left({z}\right) }\right) - \operatorname{Re} \left({f \left({z_0}\right) }\right) }\right\vert\) by Addition of Real and Imaginary Parts

It follows that:

$\forall \epsilon > 0: \exists \delta > 0: \left\vert{z - z_0}\right\vert < \delta \implies \left\vert{ \operatorname{Re} \left({ f \left({z}\right) }\right) - \operatorname{Re} \left({f \left({z_0}\right)} \right)} \right\vert < \epsilon$

Then equation $(1)$ is proven by:

\(\displaystyle \lim_{z \to z_o} \operatorname{Re} \left({ f \left({z}\right) }\right)\) \(=\) \(\displaystyle \operatorname{Re} \left({ f \left({z_0}\right) }\right)\) by definition of limit
\(\displaystyle \) \(=\) \(\displaystyle \operatorname{Re} \left({ \lim_{z \to z_o} f \left({z}\right) }\right)\) by definition of continuity


The proof for equation $(2)$ with imaginary parts follows when $\operatorname{Re}$ is replaced by $\operatorname{Im}$ in the equations above.

$\blacksquare$