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TIFR2021-Maths-B: 20

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Let $c_1,c_2>0,$ and let $f,g:\mathbb{R}\rightarrow\mathbb{R}$ be functions (not assumed to be continuous) such that for all $x\in \mathbb{R}$

$$f(x+c_1)=f(x) \text{ and } g(x+c_2)=g(x).$$

Further, assume that

$$\displaystyle \lim_{x\rightarrow\infty}(f(x)-g(x))=0.$$

Then $f=g.$

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