1. ## Continuity/Differentiability

Define a function $h : \mathbb{R} \rightarrow \mathbb{R}$ by:

(THIS IS A PIECEWISE FUNCTION):

h(x) = {0 if $x \in \mathbb{Q}$ .. x^3 + 3x^2 if $x \not\in \mathbb{Q}$

1.) Determine which points h is continuous at, and determine which points h is discontinuous at. Prove these results.

2.) Determine which points h is differentiable at, and determine which points h is non-differentiable at. Prove these results.

2. Originally Posted by fifthrapiers
Define a function $h : \mathbb{R} \rightarrow \mathbb{R}$ by:

(THIS IS A PIECEWISE FUNCTION):

h(x) = {0 if $x \in \mathbb{Q}$ .. x^3 + 3x^2 if $x \not\in \mathbb{Q}$

1.) Determine which points h is continuous at, and determine which points h is discontinuous at. Prove these results.

2.) Determine which points h is differentiable at, and determine which points h is non-differentiable at. Prove these results.
The function is continous when 0 = x^3+3x^2. Because when you approach a point x by rationals your limit is 0, when by irrationals the limit is x^3+3x^2. So need them to be equal for continuity to hold.

3. Originally Posted by ThePerfectHacker
The function is continous when 0 = x^3+3x^2. Because when you approach a point x by rationals your limit is 0, when by irrationals the limit is x^3+3x^2. So need them to be equal for continuity to hold.
Could you give me a proof for this? If it's continuous there, then it's discont. everywhere else? What about where it's diff/non-diff'able?

Thanks TPH