A function consisting of two rays and an isolated point is graphed on a coordinate plane.A function consisting of two rays and an isolated point is graphed on a coordinate plane. The first ray increases from negative infinity to the point left parenthesis negative 1 comma 2.5 right parenthesis and includes the endpoint. The ray also has a hole at left parenthesis negative 2 comma 2 right parenthesis. A single point is graphed at left parenthesis negative 2 comma 1 right parenthesis. The second ray starts at an open point at left parenthesis negative 1 comma 3 right parenthesis and decreases to infinity.

Question
Use the graph to answer the question.
Describe the continuity or discontinuity of the graphed function.
(4 points)

1 answer

To analyze the continuity or discontinuity of the function described, we can break down the information given:

  1. 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.
  2. 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.
  3. 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 \(-1\).