Math Problem Statement

Determine if f(x) is continuous at specific points (-2, 2, and 4), and from which sides (left or right) the function is continuous.

Solution

The image shows a graph of a function f(x)f(x) and asks questions about the continuity of the function at given points, including both visual and numerical parts.

Here’s how to address the questions step by step:

Part (a): Checking continuity at the given points

For each point, you must decide if f(x)f(x) is continuous:

  1. At x=2x = -2:

    • By looking at the graph at x=2x = -2, we need to check whether the function value and the limit from both sides exist and are equal.
  2. At x=2x = 2:

    • Similarly, check the function behavior at x=2x = 2 to see if it's continuous or has any jump or removable discontinuity.
  3. At x=4x = 4:

    • Check if the function behaves smoothly at x=4x = 4, ensuring the left-hand and right-hand limits match the function value.

Part (b): Determining one-sided continuity

This part requires analyzing if the function is continuous from the left, right, or both at each given point:

  1. At x=2x = -2:

    • Is f(x)f(x) continuous from the left (ll) or the right (rr)?
  2. At x=2x = 2:

    • Check whether the function is continuous from the left, right, or both at x=2x = 2.
  3. At x=4x = 4:

    • For x=4x = 4, see if the function is continuous from the left, right, or both.

Let’s summarize and provide some tips:

Would you like a detailed breakdown of each point for continuity analysis and the corresponding side continuity? I can walk through each part of the function behavior for clarity.

Related Questions:

  1. What are the conditions for a function to be continuous at a point?
  2. What is the difference between one-sided and two-sided continuity?
  3. How do you identify discontinuities on a graph?
  4. What is the concept of a removable discontinuity?
  5. Can you explain jump discontinuities and their implications?

Tip:

When determining continuity from a graph, always check the function’s value and limits from both sides at the point of interest.

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Math Problem Analysis

Mathematical Concepts

Continuity
Limits
One-Sided Continuity

Formulas

Limit of f(x) as x approaches a
Definition of Continuity at a Point

Theorems

Limit Laws
Definition of Continuity

Suitable Grade Level

Grades 10-12