# Interactive Real Analysis

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## 6. Limits, Continuity, and Differentiation

### 6.6. A Function Primer

Here we want to list some functions that illustrate more or less subtle points for continuous and differentiable functions. These functions are all difficult, in one sense or another, but should definitely be part of the repertoire of any math student with an interest in analysis.

### Examples of Continuous and Differentiable Functions Dirichlet function: A function that is not continuous at any point in R Countable discontinuities: A function that is continuous at the irrational numbers and discontinuous at the rational numbers. C1 function: A function that is differentiable, but the derivative is not continuous. Cn function: A function that is n-times differentiable, but not (n+1)-times differentiable Cinf function: A function that is not zero, infinitely often differentiable, but the n-th derivative at zero is always zero. Weierstrass function: A function that is continuous everywhere and nowhere differentiable in R. Cantor function: A continuous, non-constant, differentiable function whose derivative is zero everywhere except on a set of length zero.
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