Determinate the discontinuities of the function $f:R \to R$,$$f(x)=[x]\sin ({\pi x})$$I've seen the graph and it is clear that there are no discontinuities but how do we determine that without drawing a graph?
2026-04-01 23:43:51.1775087031
Determinate the discontinuities of the function
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If $x$ is not an integer it is clear that $f$ is continuous at $x$. If $x$ is an integer $k$ then $f(k)=0$ and $[x]\sin (\pi x) \to 0$ as $x \to k$ ( because $[x]$ remains bounded as $x \to k$ and $\sin (\pi x) \to 0$) so $f$ is continuous at $x$. Hence $f$ is continuous at all points.