• go to the website Interactive Real Analysis
• and link to Chapter 6: Limits, Continuity, and Differentiation
• then link to Section 6.6: A Function Primer.

Seven Funky Functions

1. Dirichlet function 
   • The Dirichlet Function is not continuous at any point on the real line.
   • A variant of Trench's Exercise 2.2.6.
   • Wiki page
2. Countable discontinuities 
   • A function that is continuous at the irrational numbers and discontinuous at the rational numbers.
   • Has many names, such as: Thomae's function, popcorn function, raindrop function.
   • A variant of Trench's Exercise 2.2.7.
   Wiki page
3. C¹ function 
   • A function that is differentiable, but the derivative is not continuous.
   • Excuted Maple Worksheet in HTML
   • Maple Woksheet in Maple
4. Cⁿ function  
   • A function that is n-times differentiable, but not (n+1)-times differentiable.
   • Excuted Maple Worksheet in HTML
   • Maple Woksheet in Maple
5. C^inf function 
   • A function that is not zero, infinitely often differentiable, but the n-th derivative at zero is always zero.
   • Excuted Maple Worksheet in HTML
   • Maple Woksheet in Maple
6. Weierstrass function 
   • A function that is continuous everywhere and nowhere differentiable in R.
   • A Weierstrass Function is a function of the form Σ _n=0 ^&infin   aⁿ cos (bⁿ π x)           or           Σ _n=0 ^&infin   aⁿ sin (bⁿ π x) where there are some kind of restriction on a and b.
   • A Weierstrass Function is everywhere continuous but nowhere differentiable if 0 < a < 1 and ab ≥ 1 , see (Hardy G.H., Weierstrass's nondifferentiable function, Trans - Amer. Math. Soc, 17(1916), 301-325).
   • From the Weierstrass Function's Wiki page:

     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.

   • Katie Spurrier's SCHC Senior Thesis Continuous Nowhere Differentiable Functions
   • Excuted Maple Worksheet in HTML
   • Maple Woksheet in Maple
7. Cantor function 
   • A continuous, non-constant, differentiable function whose derivative is zero everywhere except on a set of length Lebesgue measure zero.
   • The Cantor Function is a particular case of a devil's staircase.
   • The Cantor Function's Wiki page has a nice picture. Other sites with nice pictures of the Cantor Function:
     • Site 1
     • Site 2
     • Site 3
   • Here is a short article A Characterization of the Cantor Function by Donald R. Chalice.
   • Excuted Maple Worksheet in HTML    (Maple Code courtesy of I.N. Galidakis.)
   • Maple Woksheet in Maple    (Maple Code courtesy of I.N. Galidakis.)