Consider the graph of the function y=p(x) ﻿shown

in the following graph.

a. ﻿Find all values for which the function is discontinuous.

b. ﻿For each value in part a., ﻿state why the formal definition of continuity does not apply.

c. ﻿Classify each discontinuity as either jump, removable, or infinite

To analyze the continuity of the function \( y = p(x) \) based on the graph, I'll break down the steps for solving the problem:

### a. Find all values for which the function is discontinuous.

You need to look for points where:
- The function does not have a value (hole, vertical asymptote).
- The function makes a sudden jump (discontinuity in the graph).
- The function has an asymptote (where the function approaches infinity).

Please either upload the graph image, or describe the specific behavior of the graph at key points (e.g., holes, jumps, asymptotes), and I'll help identify these discontinuities.

### b. For each value in part a, state why the formal definition of continuity does not apply.

Once we've identified the discontinuities from part (a), I will check each one against the formal definition of continuity. The function \( y = p(x) \) is continuous at a point \( x = c \) if:
1. \( p(c) \) exists.
2. \( \lim_{x \to c} p(x) \) exists.
3. \( \lim_{x \to c} p(x) = p(c) \).

We'll discuss which of these criteria fail at each discontinuous point.

### c. Classify each discontinuity as either jump, removable, or infinite.

After identifying the discontinuities and why they exist, I will classify them:
- **Jump discontinuity:** The graph "jumps" from one value to another.
- **Removable discontinuity:** There's a hole in the graph where the function could be redefined to fill it.
- **Infinite discontinuity:** The function goes to infinity (vertical asymptote).

If you upload the graph or describe the points of discontinuity, I can provide specific answers to each part.

Let me know if you have the graph or can describe it!

Consider the graph of the function y=p(x) shown in the following graph. a. Find all values for which the function is discontinuous. b. For each value in part a., state why the formal definition of continuity does not apply. c. Classify each discontinuity as either jump, removable, or infinite.

Explore how to find and classify discontinuities in a function based on a graph. This problem discusses key concepts such as continuity, limits, and the types of discontinuities—jump, removable, and infinite. Ideal for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation