Continuity
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I would give you the definition of continuity, but I think it’s confusing. Plus, you should have some intuition about what it means for a graph to be continuous. Basically, a function is continuous if there are no holes, breaks, jumps, fractures, broken bones, etc. in its graph.
You can also think about it this way: A function is continuous if you can draw the entire thing without picking up your pencil. Let’s take some time to classify the most common types of discontinuity, or what makes a function not continuous.
Common Discontinuities
Jump Discontinuity
You’ll usually encounter jump discontinuities with piecewise-defined functions. “A piecewahoozle whatsit?” you ask? Exactly. A piecewise-defined function is a function for which different parts of the domain are defined by different functions. One example that’s often used to illustrate piecewise-defined functions is the cost of postage at the post office. Here’s how the cost of postage might be defined as a function, as well as the graph of this function. They tell us that the cost per ounce of any package lighter than one pound is twenty cents per ounce, that the cost of every ounce from one pound to anything less than two pounds is forty cents per ounce, etc.
The graph on the left is an example of a piecewise-defined function. Every break in this graph is a point of jump discontinuity. You can remember this by imagining yourself walking along on top of the first segment of the graph. In order to continue, you’d have to jump up to the second segment.
Point Discontinuity
Point discontinuity exists when there is a hole in the graph at one point. You usually find this kind of discontinuity when your graph is a fraction like this:
In this case, the point discontinuity exists at x=-4, where the denominator would equal zero. This function is defined and continuous everywhere, except at x=-4. The graph of a point discontinuity is easy to pick out because it looks totally normal everywhere, except for a hole at a single point.
Infinite/Essential Discontinuity
You’ll see this kind of discontinuity called both infinite discontinuity and essential discontinuity. In either case, it means that the function is discontinuous at a vertical asymptote. Vertical asymptotes are only points of discontinuity when the graph exists on both sides of the asymptote.
The first graph to the left shows a vertical asymptote that makes the graph discontinuous, because the function exists on both sides of the vertical asymptote. The vertical asymptote in the second graph on the left is not a point of discontinuity, because it doesn’t break up any part of the graph.
The Intermediate Value Theorem
Similarly to the definition of continuity, the definition of the Intermediate Value Theorem is absolutely more harmful than helpful. So instead, consider the graph to the left instead.
This theorem is fairly ridiculous because it doesn’t tell us anything that we don’t already know. All it says is that, when we look at a continuous function on a closed interval between (a,f(a)) (the blue graph) and (b,f(b)) (the purple graph), there will be a point in between them, which we’ll call (K,f(K)) (the orange graph). K must be between a and b and f(K) must be between f(a) and f(b). Looking at the graph, isn’t that obvious? Values may or may not exist below f(a) and above f(b) depending on the graph, but f(K) must exist.
