calculus - Is the function continuous - $f(x) = frac{1}{sin x} + frac{1}{x-1}$

I have an assignment in which I have to prove that a function "recieves every real value, where $x\in (0,1)$".

Here is the function:

$$f(x) = \frac{1}{\sin x} + \frac{1}{x-1}$$

I don't know the answer yet, however I wanted to know if the function is continuous. I think it is, with the following proof:

I know that if $f(x)$ and $g(x)$ are continuous then $f(x) / g(x)$ is continous as well, and the same goes for $f(x) + g(x)$.

Let $f(x)=1$ and $g(x)=\sin x$, they are both continuous and since $x \in (0,1)$, $\sin x\ne 0 $ for every value of $x$, I can say that $\frac{1}{\sin x}$ is continuous as well.

Similairly, I do this for $\frac{1}{x-1}$, and I can say the whole function is continuous.

Am I correct?

I don't know how does this make me closer to proving the argument above, but I have to start with something.

Thanks,

Alan

Answer

Your initial step is correct, $f$ is continuous on $(0,1)$. I will assume the domain of $f$ is restricted to be $(0,1)$.

Note that $\lim_{x \to 0^{+}} f(x)=\infty$ and $\lim_{x \to 1^{-}} f(x)=-\infty$, in other words near 0 and 1 the function takes arbitrarily large (or small) values.

Since the function is continuous and defined on an interval, its image is an interval (the intermediate value theorem). The only interval that contains arbitrarily large and small values is $(-\infty, \infty)$.