The Relation Between Continuity and Differentiability

Terminology: We say that a function is differentiable at a point if it has derivative there. Equivalently the graph has a non-vertical tangent line at the corresponding point on the graph. Alternatively we say that the graph is smooth at that point.

It is pretty clear that one can't draw a tangent line at a point of discontinuity of a graph. So continuity is a necessary prerequisite for differentiability. What about the converse: is continuity sufficient to insure existence of tangent lines?

To answer this question, we consider what a smooth graph looks like under a microscope. For example consider the following graph of a sine wave function. Although at first glance it looks wavy and far from being a straight line, if we zoom in on the graph with a microscope increasing the magnification as we go, it starts to flatten out and under large magnifications it looks just like a straight line. Equivalently this is how the sine curve would look to us if we were tiny bugs crawling along the graph. For the same reason the earth appears flat to us.

If you have a JAVA capable browser such as Netscape Navigator (v2.0 for Windows or v3.0 for Macintosh), you might take a look at the following interactive version of the above picture.

Now it is pretty easy to construct graphs which are continuous but nevertheless have points at which we can't draw tangent lines. A simple example is the graph of the absolute value function. The graph has a corner at the origin. No matter what magnification we apply with our microscope, we will be able to see that there is corner at the origin: it will never look anything like a straight line there.

Note that we have to keep our microscope always centered on the origin. Otherwise if we focus on some other part of the graph, then the corner will disappear from view as we increase the magnification and the graph will look like a straight line. This is why the absolute value function is differentiable away from the origin.

For a long time mathematicians thought that continuous functions had to be smooth except for some scattered bad points on the graph. Then in 1872 the German mathematician Weierstrass discovered surprising examples of functions which are continuous everywhere but nevertheless are nowhere smooth, ie. the graph has corners at all points on the graph. An example is shown here.

If you have a JAVA capable browser such as Netscape Navigator (v2.0 for Windows or v3.0 for Macintosh), you might take a look at the following interactive version of the above picture.

Weierstrass' examples were originally thought to be bizarre mathematical curiosities of no interest to science or engineering. However in the 1960's it was realized that Weierstrass' examples were the first constructions of fractals, and fractals can be used to describe a wide variety of real world phenomena. An example is the shape of a snowflake.


Here is an interactive JAVA version of the snowflake.

You might want to check out the following index of references to fractals on the world wide web: