Given a function $$f(x)=\cos\frac{1}{x}$$ Prove that it is not periodic.

I've started with assuming that $f(x)$ is periodic and hence $f(x) = f(x+2\pi)$ since $\cos$ hat period $T = 2\pi$. This gives:

$$

\cos\frac{1}{x} = \cos\frac{1}{x+2\pi}

$$

Given the above we may take some value x and check whether the equation holds for that value. Let's say $x=\frac{\pi}{2}$, then

$$

\cos\frac{2}{\pi} = \cos\frac{2}{5\pi}

$$

which is not true. Hence $f(x) = \cos\frac{1}{x}$ is not periodic. I've found similar questions here and here but they use derivatives for the proof.

Can I somehow prove the above in a general way without taking any exact value for $x$ and without using calculus?

## Best Answer

Assume $$\cos\frac1x=\cos\frac1{x+T}.$$

Then we must have

$$\frac1x=\pm\frac1{x+T}+2k\pi, $$

$$k=\frac{\dfrac1x\mp\dfrac1{x+T}}{2\pi}.$$

But the RHS cannot be an integer for all $x$, a contradiction.