Investigate the continuity of a function

I'm trying to determine whether the following function is continuous: $$y=\frac{1}{\sqrt{1+x}-\sqrt{1-x}}$$. Imo it is continuous, because it is a composition of two continuous functions $$y=\frac{1}{x}$$ and $$y=\sqrt{1+x}-\sqrt{1-x}$$.

So my answer is, that the function is continuous on it's domain and it's not continuous on $$\mathbb{R}$$. Am I correct?

Thanks

Exactly, in its domain of definition it is continuous (good exercise: try to find it). But of course not in all $$\mathbb{R}$$, since $$\frac{1}{x}$$ is defined only in $$\mathbb{R}\setminus\{0\}$$ and $$\sqrt{x}$$ makes sense only for $$x\geq 0$$.
• Thanks! So in order to investigate the continuity in $\mathbb{R}$, I have to find the domain of the composition and from there I can find easily points in which the function is not continuous? Oct 20 '19 at 12:23
• Exactly. Try first to understand when $\sqrt{1+x}-\sqrt{1-x}$ makes sense, and in order to compose it with $\frac{1}{x}$, when it it equal to zero. Then, intersect the two conditions on $x$ to obtain the domain of definition. Oct 20 '19 at 12:27
• To be more precise, rewrite the function multiplying all by $\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}$ in order to get rid of the square roots at the denominator. Oct 20 '19 at 12:29
You are correct but a more detailed answer would explain the domain as well. The domain is $$[-\sqrt 2 /2,0)\cup (0, \sqrt 2/2]$$