Let [t] denote the greatest integer t and _x0x[4x]=A.

Question

Let $[t]$ denote the greatest integer $\le t$ and $\displaystyle\lim_{x	o0}x\left[\dfrac4xight]=A$. Then the function $f(x)=[x^2]\sin(\pi x)$ is discontinuous, when $x$ is equal to:

Accepted Answer

$\displaystyle\lim_{x	o0}x\left[\frac4xight]=\lim_{x	o0}\left(4-x\left\{\frac4xight\}ight)=4$, so $A=4$.

$f(x)=[x^2]\sin\pi x$ is discontinuous wherever $x^2$ is a non-negative integer but $x$ is not an integer (so that $\sin\pi x
e0$ at the jump). $x=\sqrt{A+1}=\sqrt5$, which is not an integer, so $f$ is discontinuous there.

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