Exercise 4.16

Question

Exercise 16: Let $[x]$ denote the largest integer contained in $x$ and let $(x)=x-[x]$ denote the fractional part of $x$. What discontinuities do the functions $[x]$
    and $(x)$ have?

analambanomenos · Accepted Answer

Let $f(x)=[x]$ and $g(x)=(x)$. Then $f$ and $g$, being constant or linear functions on $(n,n+1)$ for each integer $n$, are continuous except at the integers. If $n$ is an
    integer, then $f(n-)=n-1$ and $f(n+)=f(n)=n$, and $g(n-)=1$ and $g(n+)=g(n)=0$. These are all simple discontinuities.

Exercise 4.16

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Exercise 4.16

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