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I have to explain whether the piece-wise function below has any removable discontinuities. I am confused because, as far as I know, to determine whether there is a removable discontinuity, you need to have a mathematical function, not simply a condition.

Is there some way I could tell whether the function below has any removable discontinuities?

$$f(x)=\left\{\begin{array}{ll} {1,} & {x \leq -1} \\ {-x,} & {-1 <x<0} \\ {1,} & {x=0} \\ {-x,} & {0<x<1} \\ {1,} & {x \ge 1}\end{array}\right.$$

Graph of Function:

enter image description here

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    $\begingroup$ What is the definition of a removable discontinuity? From that, your graph (crude or not) should readily provide the answer. $\endgroup$ May 9, 2019 at 20:44
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    $\begingroup$ By the way this is the graph in Desmos - desmos.com/calculator/pvw7k8is42 $\endgroup$ May 9, 2019 at 20:54
  • $\begingroup$ @PeterForeman Thanks! Now I can make my own! $\endgroup$ May 9, 2019 at 20:55

1 Answer 1

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Even though it is written as a condition, $f(x)$ is in fact a function.

To find a removable discontinuity, we want to have a point such that $$\lim_{x\rightarrow x_0+} f(x) =\lim_{x\rightarrow x_0-} f(x) \neq f(x_0)$$ where the limits are those from the left and the right.

Note that at $x=0$, we have that the limit approaching $0$ from the right of $f$ and the limit approaching $0$ from the left of $f$ are both $0$ but $f(0)=1$. Therefore there is a removable discontinuity at $x = 0$.

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