Differentiable / continuous functions

[jem05]

Homework Statement

give an example of a function f: R --> R that is differentiable n times at 0, and discontinous everywhere else.

Homework Equations

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The Attempt at a Solution

i got one, and i proved everything, i just want to make sure what i did is correct:

f:x ⁿ⁺¹ when x is rational
0 when x is irrational

by the way, does the example hold if i invert them, that is 0 if rational and xⁿ⁺¹ if irrational?
(nothing changes right?)
thank you

oh, and x^n does not work, instead of x^n+1, right?

 

[Hootenanny]

Homework Statement

give an example of a function f: R --> R that is differentiable n times at 0, and discontinous everywhere else.

Homework Equations

---

The Attempt at a Solution

i got one, and i proved everything, i just want to make sure what i did is correct:

f:x ⁿ when x is rational
0 when x is irrational

btw, does the example hold if i invert them, that is 0 if rational and xⁿ if irrational? (nothing changes right?)
thank you

Your function is differentiable (and hence continuous) everywhere on R, except at a finite number of points. Indeed, your function is differentiable on at least $\mathbb{R}\setminus\mathbb{Q}$.