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.