The intuitive idea behind the mathematical concept of continuity is that a function is continuous if small changes in the input produce small changes in the output. If small changes in the input can produce abrupt changes in the output, the function is said to be discontinuous.
As an example, consider the function h(t) which describes the height of a growing child at time t. This function is continuous: the height doesn't "jump". If M(t) denotes the amount of money in your bank account, then this function is discontinuous: if you take money out, the function jumps abruptly to a new value.
Mathematically, the concept is defined as follows. First consider the case of a function f that maps a set of real numbers to another set of real numbers, like the two examples h and M above. This function f is said to be continuous at the point x=x0 if (and only if) the following holds: For any positive number ε however small, there exists some positive number δ such that for all x with x0 - δ < x < x0 + δ, the value of f(x) will satisfy f(x0) - ε < f(x) < f(x0) + ε. More intuitively, we can say that if we want to get all the f(x) values to stay in some teeny neighborhood around f(x0), we simply need to choose a small enough neighborhood for the x values around x0, and we can do that no matter how teeny the f(x) neighborhood is.
If f is continuous at every point of its domain, then we call it continuous, otherwise discontinuous.
An example of a discontinuous function is the function f defined by f(x) = 1 if x > 0, f(x) = 0 if x <= 0. There is no δ-neighborhood around x=0 that will force all the f(x) values to be within ε of f(0), for any ε < 1. Intuitively we can think of a discontinuity as a sudden jump in function values.
Continuity can also be characterized using limits: A function f is continuous at x=x0 if and only if limx->x0 f(x) = f(x0).
To generalize, consider a function f from one metric space (X, dX) to another metric space (Y, dY). Then f is continuous at the point x0 in X if for any positive real number ε, there exists a positive real number δ such that all x in X satisfying dX(x, x0) < δ will also satisfy dY(f(x), y0) < ε.
More generally still, we can define continuity for functions between topological spaces. Suppose f is a function from a topological space X into a topological space Y. Then f is said to be continuous at a point x in X if for every neighbourhood V of f(x) there is a neighbourhood U of x such that f(U) is a subset of V. If f is continuous at every point of X, then it is simply said to be continuous. It turns out that a function is continuous if and only if the pre-image of every open set is open, and so this is often used as the definition of continuity.