This section describes one of the most important concepts in calculus - continuity!
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Continuity is often described as the ability to draw a graph “without picking up your pencil”. What we really mean by this, is that there are no gaps in the curve - which are called discontinuities (which we will discuss in a future segment). But there are still many variations of how a function can be continuous. In this spirit, consider the following graph:
This function looks kind of crazy, but notice that it is still continuous - indeed, there are no gaps anywhere along the curve. Moreover, there is something else noteworthy about this graph... it’s smooth. This seems like a subjective term, but in mathematics it has real meaning (again the analytic nuts and bolts are beyond the scope of this course, but we can get a feel for it visually!) Consider the following graph for contrast:
At first glance this graph is similar to our other one, but it looks... jagged. Indeed, it has all kinds of points where the graph rapidly shifts direction - so rapidly in fact that it creates a sharp corner. This is an example of a “not smooth” function - in particular a “corner” is an example of something that is continuous, but not smooth. For another example, consider the following graph:
This is clearly continuous still, but it seems a little more spiky than a corner. Indeed, this kind of point is called a cusp, and is the other classic example of a graph that is continuous but not smooth.
Corners and cusps are the classic examples of continuous but not smooth points, and are studied in greater detail in calculus. Nonetheless, it is worth being aware that just because we know a function is continuous, it doesn’t necessarily mean that the function is “nice” - it might still be jagged and spiky, but it’s at least all still nicely connected without any gaps.