Using the continuity concept, it is found that the function is continuous for all real values.
What is the continuity concept?
A function f(x) is continuous at x = a if it is defined at x = a, and:
[tex]\lim_{x \rightarrow a^-} f(x) = \lim_{x \rightarrow a^+} f(x) = f(a)[/tex]
In this problem, we are given a piece-wise function, hence we have to look at the points where the definition of the function changes. In this problem, it can only be discontinuous at x = 0, which we have to verify.
Then:
- [tex]f(0) = 0^2 - 1 = -1[/tex].
- [tex]\lim_{x \rightarrow 0^-} = \lim_{x \rightarrow 0} x^2 - 1 = 0^2 - 1 = -1[/tex].
- [tex]\lim_{x \rightarrow 0^+} = \lim_{x \rightarrow 0} \frac{1}{x - 1} = \frac{1}{0 - 1} = -1[/tex].
The 3 values are equal, hence the function is continuous at x = 0 and for all real values.
More can be learned about the continuity of a function at https://brainly.com/question/24637240
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