To analyze the continuity or discontinuity of the function described, we can break down the information given:
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First Ray:
- It increases from negative infinity to the point \((-1, 2.5)\), and this endpoint is included in the function, which means the function is continuous up to this point.
- There is a hole at \((-2, 2)\), which signifies a discontinuity since the function is not defined at this point. This means that the limit of the function as it approaches \(-2\) does not equal the function value at that point.
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Isolated Point:
- There is an isolated point at \((-2, 1)\). This point does not connect to the hole at \((-2, 2)\), indicating that at \(-2\), the function jumps from not existing at the hole to being \(1\) at the isolated point. This creates a discontinuity as well.
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Second Ray:
- The second ray starts at an open point at \((-1, 3)\) and decreases to infinity. The open point means that the function is not defined at \((-1, 3)\), which contributes to the discontinuity since the function does not approach this point from the left (it comes from the ray).
Summary of Continuity or Discontinuity:
- The function has a discontinuity at \(-2\) due to the hole and the isolated point.
- The function is also discontinuous at \(-1\) because of the open point there.
- The transition from the first ray to the second ray creates additional points of discontinuity at \(-1\).
Final Description:
The graphed function is discontinuous at \(-2\) (due to the hole and isolated point) and at \(-1\) (due to the open point), while being continuous on the intervals defined by those points outside these discontinuities. Overall, the function exhibits a jump discontinuity at both \(-2\) and \(-1\).