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**Enrique2** Be careful, you need a uniform bound for $\displaystyle 1/x^2$ for applying M-test. Your argument is correct for the compact subsets of $\displaystyle (-\infty,0)\cup (0,\infty)$, when such a bound always exists.

Just observe that this gives you an answer for (d), for each

$\displaystyle x\neq 0$ you have that there exists $\displaystyle x\in[a,b]\subseteq \mathbb{R}\setminus\{0\}$ such that the series converges uniformly on $\displaystyle [a,b]$.

Now, recall that the uniform convergence of continuous functions preserves continuity. Actually M-test gives continuity whenever the terms of the series are.