Is $\cos(x[n])$ stable? I am highly confused about the stability of $\cos(x[n])$.

If we provide a bounded input such as $x[n]=u[n]$, the output is bounded.

Now if we provide an unbounded signal $x[n]=\delta[n]$, the output oscillates between $-1$ and $1$. So I think it's marginally stable in this case.

Am I correct? I have this confusion because in my book it says it's unstable when $x[n]=\delta[n]$. So my questions are: Is the function BIBO stable and also is it stable for an unbounded input ?

Note: $u[n]$ and $\delta[n]$ are unit step and dirac delta functions respectively.



Correction: As mentioned in the comment, $x[n]=\delta[n]$ is bounded too.

But this points more strongly to $\cos(x[n])$ being stable, isn't it?
The I/O relation
$$y[n] = \cos(x[n])$$

represents a stable system. 

Because by definition of $\cos(x)$, irrespective of the input $x[n]$, its output will always be bounded by $$-1 \leq \cos(x[n]) \leq 1 $$ for all $x[n]$ for all $n$. Even if $x[n]$ is undefined ( such as would be the case with $x[n] = 1/n$ at $n=0$ as Marcus pointed out) we for sure know that the system output is bounded (the value of input is undefined, but the output is known to be within $\pm1$.)