Limits and Continuity in ℂ

A complex function has limits, just like a real one — but the geometry is richer. We write

\lim_{z \to z_0} f(z) = L

to mean that f(z) gets arbitrarily close to L as z gets close to z_0. In terms of distances in the plane:

|f(z) - L| \to 0 \quad\text{whenever}\quad |z - z_0| \to 0.

Both |z - z_0| and |f(z) - L| are ordinary moduli — real, non-negative distances — so a complex limit is really a statement about two shrinking lengths.

Approach from every direction

Here is the crucial difference from the real line. On the reals, a point can only be approached from the left or the right — two directions. In the plane, z can slide toward z_0 along infinitely many paths: straight in from any angle, in along a spiral, in along a wiggling curve.

For the limit to exist, f(z) must approach the same value L no matter which path is taken. If two different paths give two different destinations, the limit simply does not exist.

Step 1 — fix the target. Pick z_0 and a candidate value L.

Step 2 — demand path-independence. Every way of letting z \to z_0 must drive |f(z) - L| to 0. This single requirement, harmless-looking now, is the seed of everything that makes complex differentiation so powerful.

Continuity

A function is continuous at z_0 when the limit exists and equals the value there — no jump, no hole:

\lim_{z \to z_0} f(z) = f(z_0).

Step 1 — the limit must exist (the same L along every path).

Step 2 — it must hit the value. That common L has to equal f(z_0) exactly.

All the usual reassurances carry over: sums, products, and quotients (away from zeros of the denominator) of continuous functions are continuous, so every polynomial in z is continuous on the whole plane. For example

\lim_{z \to 2 + i} z^2 = (2 + i)^2 = 4 + 4i + i^2 = 3 + 4i,

which is just f(2 + i) — squaring is continuous, so the limit is found by plugging in.

For a complex function f:

\lim_{z \to z_0} f(z) = L means |f(z) - L| \to 0 as |z - z_0| \to 0;
the limit must give the same L along every path of approach — infinitely many, not just two;
f is continuous at z_0 iff \lim_{z \to z_0} f(z) = f(z_0); polynomials are continuous everywhere.

On the real line, a one-sided pair of limits is enough to settle existence — there are only two ways in. In the plane you give that up: a function can sail smoothly into z_0 along the real axis, smoothly along the imaginary axis, and still disagree between the two. That tension — every direction must conspire to the same answer — is what the next pages exploit. When we ask the difference quotient to have a single limit from all directions, we are imposing an enormous, almost rigid constraint on f.

Slide in from any angle

Fix the target z_0 at the centre. The angle slider chooses a direction of approach, and the distance slider sets how far out z starts. Push distance toward 0 and watch |z - z_0| — and the image gap |f(z) - L| for f(z) = z^2 — both shrink. For a continuous function, every angle drives that gap to zero.