# How to determine whether a piecewise function with conditions instead of equations has removable discontinuities?

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

Graph of Function:

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

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 $$x = 0$$.