Open any calculus book and you will find a discussion about how differentiability implies continuity, but continuity does not imply differentiability. The absolute value function is the standard example of a continuous function that is not differentiable (at ).
The inquisitive student may ask: how bad can continuous, nondifferentiable functions get? Can we make a function nondifferentiable at an infinitely many points? At a dense collection of points? At every point?
This is the topic in the Real Analysis class I’m teaching right now. Surprisingly, there are functions that are continuous everywhere, but differentiable nowhere! More surprisingly, it is possible to give an explicit formula for such a function.
Weierstrass was the first to publish an example of such a function (1872). It appears that Bolzano had an example as early as 1830, but it wasn’t published until much later.
The example I gave in class is called the Blancmange function (and the graph is called the Takagi fractal curve).
Let be the sawtooth function equal to on and repeated periodically elsewhere.
Then , the graph of which is shown below, is a continuous and nowhere differentiable. I created a Geogebra applet to help the students visualize the construction of this function.
Although this seems like a crazy example, it turns out that most continuous functions are nowhere differentiable (there is a technical meaning for “most”). The nice functions that we see every day in our calculus classes are rare.