Given a function $f:\Bbb R\to\Bbb R$ defined as $f(x)=2x+[x]+\text{sin}x\text{cos}x$, I need to confirm if it's one to one and onto.
I checked for the monotonicity by differentiating the function to get $f'(x)=2+cos2x>0$ (here I took differentiation of $[x]=0$) and confirmed that it's increasing since $\text{cos}2x$ has a range of $[-1,1]$ and hence is one one .
How do I check for onto here? Is it valid to say that since $[x]$ is discontinuous my $f(xn)$ will also be discontinuous, I am not sure if this statement is always correct, so please explain this or some other method to explain why it will be an onto function.
Best Answer
For one-to-one by monotonicity, you just have to complete your "$f′>0$ at non-integer points" with "the jump of $f$ at integer points is $>0$".
For (non!-) surjectivity, look at the link (given in comments) to WolframAlpha's plot. Or simply realize that $f$ is increasing and has these positive jumps.