In single variable calculus of reals, the definition of a limit is split into two cases, the limit from above and the limit from below.

When defining a derivative in complex analysis, we use the same definition as we do in single variable calculus, namely, that f'(z) = (f(z + delta z) – f(z) )/(delta z)

However, now, delta z is allowed to approach z in any direction, and actually along any curve. This is a much stronger condition the having the limit approach from both sides, in the case of a real variable.

It turns out we also need to be careful, as the limit may not exist. For example, if f(z) = z conjugate, then f'(0) is one value when taking the limit along the positive real axis, and another thing when taking the limit along the positive imaginary axis. (Try it!).