What does it mean that a given function is continuous - and how can the continuity of a function be determined?
Defining Continuity
Mathematical continuity refers to a graph in which a function can be "followed" along by a pen or pencil through a given point without having to lift the writing utensil off the paper.
Take, for example, the function f(x) = x. Anyone familiar with basic calculus will recognize this as the graph of a simple line which passes through the origin - point (0,0) - at a forty-five degree angle.
A common, simple question which might be asked in calculus is whether or not this function is continuous for a certain x-value. For instance, one might be asked: Is f(x)=x continuous at x=3? All this is asking is whether or not the graph of the function extends to, through, and away from x=3 in a continuous manner. In other words, the very term "continuous" is incredibly descriptive of what is being referred to here.
What is a non-continuous function? Well, the function above could be made non-continuous simply by adding the limitation: f(x)=0 when x=3. This would change the function in such a way that a line could no longer be drawn directly through the point x=3 continuously. At that point, there would be a hole in the graph, and continuity would be lost.
In a much more mathematical sense, one could say that the continuity of a function is dependent upon three key points:
1) The limit at the point in question must exist (both the left-hand and right-hand limit, not just one or the other) - See the Suite101 article on Limits to better understand what this means.
2) The limit to the function must be equal to the point in question. In other words, if one is curious about the continuity of a function at point a, then the limit of f(x) as x approaches a must equal f(a).
3) The point in question must be defined. Simply put, it has to exist in the first place
Using Continuity
Why is continuity important in calculus?
Determining the continuity of a function (and there are a great many tricks to doing this) allows a mathematician an integral piece of knowledge which can be used to take calculus to even further levels of complexity. For instance, it is only by determining the continuity of a function of at a given location which allows the function to be differentiated at that place.
The key principle of calculus, after all, is to analyze a given function and to find out as much about it as possible (it is for this reason that calculus is often referred to simply as "analysis").
Continuity also finds importance outside of calculus, such as in topology, where geometrical surfaces and figures are analyzed rather than functions.
Continuity also makes its way into the physical sciences to a surprising extent. In fact, a great portion of modern theoretical physics revolves around the question of continuity - about whether or not the universe itself is defined by continuous or discontinuous values.
From just the simplest definition - A function which can be drawn through a point without lifting pen from paper - a great number of exciting and important conclusions can be drawn, and one swings open the doors to reveal an entire host of mathematical and scientific questions.
This is one of the exciting aspects of mathematics.
