Define [tex]h : \mathbb{R} \rightarrow \mathbb{R}[/tex](adsbygoogle = window.adsbygoogle || []).push({});

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

a.) Determine at what points [tex]h[/tex] is continuous and discontinuous. Prove results.

b.) Determine at what points [tex]h[/tex] is differentiable and non-diff'able. Prove results.

My work:

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

Not sure how to prove this.

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

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# Continuity with the following function

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