Prove that function `f(x)=cos sqrt(x)` is non-periodic.

Explanation

We have `f(x)=cos sqrt(x)`
Let f(x) be periodic with period T, where `T gt 0.`
` :. f(x+T)=f(x)`
`implies cos sqrt(x+T)=cos sqrt(x) " for " x ge 0.`
In particular choosing ` x=0`, we have
`cos sqrt(T)=cos sqrt(0)=1 " …(1)" `
For `x=T`, we have
`cos sqrt(T+T)=cos sqrt(T)=1`
or `cos sqrt(2T)=1 " ...(2)" `
From (1) , `sqrt(T) =2m pi, m in Z`
From (2), `sqrt(2T)=2n pi, n in Z`
` :. (sqrt(2T))/(sqrt(T))=(2n pi)/(2m pi)`
or ` sqrt(2)=(n)/(m),` which is not true.
So, `cos sqrt(x)` is not periodic.