Discuss the continuity of the function defined by $$f(x)=\left\{\begin{array}{r} x+2, \text { if } x<0 \\ -x+2, \text { if } x>0 \end{array}\right.$$

Step 1: Identify the critical point

The function $f(x)$ is defined piecewise, with different expressions for $x<0$ and $x>0$. The point where the definition changes is $x=0$. We need to check the continuity of the function at this critical point.

Step 2: Calculate the left-hand limit

To find the left-hand limit as $x$ approaches $0$, we use the function definition for $x<0$, which is $f(x) = x+2$. Substituting $x=0$ into this expression gives us the left-hand limit.

Step 3: Calculate the right-hand limit

To find the right-hand limit as $x$ approaches $0$, we use the function definition for $x>0$, which is $f(x) = -x+2$. Substituting $x=0$ into this expression gives us the right-hand limit.

Step 4: Check the function value at $x=0$

The function $f(x)$ is defined as $x+2$ for $x<0$ and $-x+2$ for $x>0$. There is no explicit definition for $f(x)$ at $x=0$. Therefore, the function is undefined at $x=0$.

Step 5: Determine continuity at $x=0$

For a function to be continuous at a point, the left-hand limit, the right-hand limit, and the function value at that point must all exist and be equal. In this case, while the left-hand limit and right-hand limit are equal, the function value at $x=0$ is undefined. Therefore, the function is discontinuous at $x=0$.

Step 6: Determine continuity for x < 0 and x > 0

For $x<0$, $f(x) = x+2$, which is a polynomial function. Polynomial functions are continuous everywhere. Similarly, for $x>0$, $f(x) = -x+2$, which is also a polynomial function and thus continuous everywhere. Therefore, the function is continuous for all $x \in \mathbb{R}, x \neq 0$.

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