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2026-04-05 23:06:05.1775430365
Prove that a function similar to Dirichlet is continuous under some condition.
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Hint :
Use sequential criterion to show $f$ is continuous exactly at where $\sin\vert x \vert\equiv0 $ .
Added : Let $x \neq k \pi, k \in \Bbb{Z}$
If $\{x_n\} \in \Bbb{Q}$ such that $x_n \rightarrow x$, then by continuity of sine function, $$\lim f(x_n)=\lim \sin \vert x_n \vert=\sin (\lim \vert x_n \vert )=\sin \vert x \vert \neq 0$$ Whereas if $\{x_n\} \in \Bbb{Q}^c$ such that $x_n \rightarrow x$, then $$\lim f(x_n)=0$$ so $f$ is discontinuous at $x \neq k \pi, k \in \Bbb{Z}$