Check if $f(x)=2 [x]+\cos x$ is many-one and into or not? If $f(x)=2 [x]+\cos x$ 
Then $f:R \to R$ is: 
$(A)$ One-One and onto
$(B)$ One-One and into
$(C)$ Many-One and into
$(D)$ Many-One and onto
$[ .]$ represent floor function (also known as greatest integer function
)
Clearly $f(x)$ is into as $2[x]$ is an even integer and $f(x)$ will not be able to achieve every real number.
Answer says option$(C)$ is correct but I cannot see $f(x)$ to be many-one as it does not look like that value of $f(x)$ is same for any two values of $x$
e.g. $f(x)= [x]+\cos x$, then $f(0)=f(\frac{\pi}{2})=1$ making the function many-one but can't see it happening for $f(x)= 2[x]+\cos x$
Could someone help me with this?

You are right with respect to surjectiveness (it is not onto).
Hint:
For injectiveness (one to one), look in a neighbourhood around $x = 3\pi$ for example.
