Monotony of $f(x)=x+x^2\sin(\frac{1}{x})+1$ (Prove $f$ is strictly increasing)

Question

Let $f(x)=x+x^2\sin(\frac{1}{x})+1 , x \neq 0$ and $f(0)=1$.I am asked to find the $f(A)$.

We can easily show that $\lim_{x\to -\infty}f(x) = -\infty$ and $\lim_{x\to \infty}f(x) = \infty$. I also know that f is continuous. Now I am must prove that f is strictly monotonic, and in fact strictly increasing ( So it goes from $-\infty$ to $+\infty$).

For $ f'(x)= 1 + 2x\sin(\frac{1}{x})- \cos(\frac{1}{x})$. I don't think this can prove that however.Do I go for $f''$?Can anyone suggest anything?

Answer 1

If we define $f(0):=1$ , then $f$ is continuous on $ \mathbb R.$

Since $ \lim_{x \to \pm \infty}f(x)= \pm \infty,$ the intermediate value theorem gives $f( \mathbb R) = \mathbb R.$

Answer 2

The following graph shows that the function extended to $x=0$ with $f(0)=1$ is continuous, and that the title question is impossible (if the domain is a neighbourhood of zero):