The concept of continuous functions appears everywhere. All of calculus is about them. In fact, calculus was born because there was a need to describe and study two things that we consider "continuous": change and motion.
In calculus, something being continuous has the same meaning as in everyday use. For example, the growth of a plant is continuous. It doesn't grow by leaps, but continuously.
We know intuitively when something is continuous. For example, I'm pretty sure you'll agree that the following things are continuous:
And I hope you'll agree the following things are not continuous:
The opposite of something continuous (using this everyday usage of the word continuity) is called discrete.
Here we are talking about the physical intuition behind continuous functions. But how we translate this intuitive concept into the language of math, specifically, that of functions?
As in many other instances, here graphs can help a lot to develop a clear intuitive understanding. Here are some graphs of continuous functions:
Intuitively, we can say that a function is continuous when you can draw its graph without lifting your pencil. So, clearly, the functions above are continuous.
Now, let's see an example of a discontinuous function:
Clearly, you can't draw this graph without lifting your pencil at some point. This intuitive definition of continuous functions is easy to understand, but it is not specific enough. How could a computer decide whether or not a given function is continuous, using that definition?
To reach a more exact definition of a continuous function, we first need a slightly modified concept of limit.
When we talked about limits, we said that the expression:
Means that as x approaches a, f(x) approaches L. Graphically:
Here, it doesn't matter "how" x approaches a. It could approach it with values greater than a, or values that are less than a.
If we only allow x to approach a with values that are less than a, we say that we are taking the left-sided (or left-handed) limit, and it is expressed as:
Conversely, if we only allow x to approach a with values greater than a, we say that we are taking the right-sided (or right-handed) limit, and we write it as:
This expression is read: "The limit of f(x) as x approaches a from the right equals L".
For example, in the function:
The left-sided limit as x approaches a is:
And the right sided limit is:
We can see (using our intuitive definition) that this function is not continuous. Why? Because it has a leap. In terms of one-sided limits, this is expressed as:
So, our first condition for a function to be continuous is that at each point the one-sided limits must be equal.
Is this sufficient to cover all cases of discontinuous functions? What about the following function:
We can see that:
But, this function is not continuous at point a. To cover this case we must include the condition:
But there's a nice detail here. Whenever we have that the one-sided limits are equal, we also have that they equal the "common" limit, that is:
So, we can reduce our conditions for a function to be continuous at a point "a" to a single one:
In most textbooks they tell you that for a continuous functions, two conditions are met:
The second condition is what we saw in the previous section. The first one, though, I believe, is nonsense.
If f(x) is not defined for x=a, we can't even talk about f(x) being continuous at that point. Why? Because continuity is only defined for points in the domain of the function!
As we saw in domain of a function, for a function to be correctly defined, we must specify the domain. For example:
This function is not defined for x=1. So, the domain must be all real numbers that aren't 1. We can't even talk about the continuity of this function at x=1.
However, if the define the piecewise function:
We have that, if x is distinct from 1:
But f(1)=0. So, we have that:
This implies that this is not a continuous function. It looks like this:
Most "discontinuous" functions you'll encounter in problems will be like the previous one (not the piecewise function, but the first one). Your job would be to find (if it exists) a point where the function is not defined.