MHB Find Discontinuation Points of f(x) = x*[x]

[Yankel]

Hello all.

I am trying to find the discontinuation points of the function: f(x) = x*[x]

I have a solution attached, according to which the function is not continuous for every x in Z, apart from x=0, and continuous for every x not in Z and for x=0. However, trying to plot this in maple gave different results. While this was the answer for x*ceil(x) or x*floor(x), for x*round(x) I got a plot that doesn't match the answer. I want to ask you, can you please explain to me what [x] means and where is the function x*[x] continuous?

Thank you

 

[Deveno]

I am unsure what $[x]$ means, as well-it's possible it may mean the "fractional part" of $x$, that is (for example):

$[\pi] = \pi - 3$ (to be more precise $[x] = x - \lfloor x \rfloor$ if $x \geq 0$, and $[x] = x - \lceil x \rceil$ for $x < 0$).

I believe it is easiest to approach this problem (if I am correct about what $[x]$ means) with a two-pronged attack:

1. Show that if $x \not\in \Bbb Z$, that $[x]$ is continuous (on some interval containing $x$), and thus $f$ is continuous.

2. Show if $x \in \Bbb Z - \{0\}$ that $f$ is discontinuous directly from the definition ($\epsilon-\delta$ criterion).

$x = 0$ is a "special case", because of the factor $x$ in $f(x) = x[x]$.

It would help if we had an image of your function.