continuity of a complex function
A complex function f is continuous at a point z_0 if its value there is exactly what the nearby values are heading toward — no jump, no hole. In symbols, f is continuous at z_0 when the limit of f(z) as z tends to z_0 exists and equals f(z_0). Intuitively, if you nudge the input z a tiny bit, the output f(z) only moves a tiny bit; the function does not tear the plane apart.
Spelled out with epsilons: f is continuous at z_0 if for every epsilon > 0 there is a delta > 0 such that |z - z_0| < delta forces |f(z) - f(z_0)| < epsilon. A function is continuous on a set if it is continuous at every point of that set. Because continuity is built on the complex limit, it inherits the all-directions demand: f must approach f(z_0) along every route into z_0, not just along the axes.
Continuity behaves exactly as you would hope under combining functions, which makes most functions you write down continuous without any fuss. Sums, differences, and products of continuous functions are continuous; a quotient f/g is continuous wherever g is non-zero; and a composition of continuous functions is continuous. Polynomials in z are continuous everywhere, and rational functions are continuous everywhere except where the denominator vanishes. Continuity is the gentle baseline; the dramatic strength of complex analysis comes one level up, at differentiability.
f(z) = z^2 + 1 is continuous everywhere (it is a polynomial). The function g(z) = 1/(z - i) is continuous everywhere except z = i, where its denominator vanishes and the output blows up.
Polynomials are continuous everywhere; rational functions, everywhere their denominator is non-zero.
Continuity is far weaker than complex-differentiability. z-bar and |z| are continuous everywhere yet nowhere holomorphic. Continuity only forbids jumps and holes; it says nothing about the rigid Cauchy-Riemann structure that differentiability demands.