Continuity with the following function

[Caeder]

Define $$h : \mathbb{R} \rightarrow \mathbb{R}$$

$$h(x) = \begin{cases} 0 &\text{if\ }\ x \in \mathbb{Q}\\ x^3 + 3x^2 &\text{if\ }\ x \notin \mathbb{Q} \end{cases}$$.



a.) Determine at what points $$h$$ is continuous and discontinuous. Prove results.

b.) Determine at what points $$h$$ is differentiable and non-diff'able. Prove results.

My work:

$$h$$ is obviously cont. when $$0=x^3 + 3x^2$$ as when approaching x, lim is 0 by rationals and lim is $$x^3 + 3x^2$$ by irrationals. So, it's cont. at $$x=0,-3$$ and discont. everywhere else.

Not sure how to prove this.

And for differentiability.. not sure.. diff'able nowhere?

 

[mathwonk]

maybe diffble at 0?

 

[SiddharthM]

use sequences to prove (a), remember the theorem that f is continuous on a set E if for every sequence x_n in E that converges to x_0 in E has the property that f(x_n) converges to f(x_0).

for (b) u can use sequences to find out where lim[(f(x)-f(y))/(x-y)] as x goes to y, where y is fixed exists.