| Example 1: | A function that is not continuous at any point in R. |
| Example 2: | A function that is continuous at the irrational numbers and discontinuous at the rational numbers. |
| Example 3: | A function that is differentiable, but the derivative is not continuous. |
| Example 4: | A function that is n-times differentiable, but not (n+1)-times differentiable. |
| Example 5: | A function that is not zero, infinitely often differentiable, but the n-th derivative at zero is always zero. |
| Example 6: | A function that is continuous everywhere and nowhere differentiable in R. |
| Example 7: |
A continuous, non-constant, differentiable function
whose derivative is zero everywhere except on a set of
|
Example 2 No plugins. This is Exmple 4.2.2g from our textbook.
Example 3 Plugins. See homework problem 5.1.13.
Example 4 Plugins.
Example 5 Plugins. Example 6 Plugins. Example 8.5.3 in our textbook. A Weierstrass Function is a function of the form| It turns out that the Weierstrass function is far from being an isolated example: although it is "pathological", it is also "typical" of continuous functions. In a topological sense, it can be shown that the set of nowhere-differentiable real-valued functions on [0, 1] is dense in the vector space C([0, 1]; R) of all continuous real-valued functions on [0, 1] with the topology of uniform convergence. |
Example 7 No Plugins but also no graphics. The example is the Cantor Function, which we have talked about in class. It is a particular case of a devil's staircase.