A property of the function $\frac{\sin x}{x}$

Let $0\ne r\in [-1,1].$ Let $f_r(x)=-rx+\sin x.$

Since $\frac {\sin x}{x}\to 0$ as $|x|\to \infty,$ take $M>0 $ such that $|x|>M\implies \left|\frac {\sin x}{x}\right|<|r|\implies f_r(x)\ne 0.$

Now $f_r'(x)=-r+\cos x$ so the set $S=\{x\in [-M,M]: f'_r(x)=0\}$ is finite. So let $S\cup \{-M,M\}=\{x_j: 1\le j\le n\}$ for some $n\in \Bbb N,$ where $x_j<x_{j+1}$ for each $j<n.$

Now $f_r$ is strictly monotonic on each interval $[x_j,x_{j+1}]$ for $j<n$ because $f'_r$ is continuous and non-$0$ on $(x_j,x_{j+1}).$ So there is at most one $x\in [x_j,x_{j+1}]$ such that $f_r(x)=0.$

We could also say there is a member of $(f'_r)^{-1}\{0\}$ between any 2 members of $f_r^{-1}\{0\}$ so if $[-M,M]\cap f_r^{-1}\{0\}$ was infinite then $[-M,M]\cap (f_r')^{-1}\{0\}$ would be infinite.