Show that $x+a-[x-b],a,b\in\mathbb R^+$, is a periodic function. Find its period. $([.]$ represents greatest integer function.)

Question

Show that $x+a-[x-b],a,b\in\mathbb R^+$, is a periodic function. Find its period. $([.]$ represents greatest integer function.)

My Attempt: Replacing $x$ by $[x]+\{x\}$, where $\{x\}$ is a fractional part of $x$.

$[x]+\{x\}+a-[[x]+\{x\}-b]=[x]+\{x\}+a-[x]-[\{x\}-b]=\{x\}+a-[\{x\}-b]$

How to proceed next?

Answer 1

We know that $[x]=x-\{x\}$, so \begin{align*} x-a-[x-b]&=x-a-(x-b-\{x-b\})=x-a-x+b+\{x-b\}\\ &=b-a+\{x-b\}. \end{align*} Since $\{x\}$ has period 1, our function also has period 1

Answer 2

Let $f(x) = x + a - [x - b].$

Note that $[x + 1 - b] = 1 + [x-b].$

Therefore, $f(x+1) = (x + a + 1) - (1 + [x-b]) = f(x).$