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1.12 confirming continuity over an interval

1.12 confirming continuity over an interval

3 min read 10-12-2024
1.12 confirming continuity over an interval

Meta Description: Learn how to rigorously prove continuity of a function over a given interval. This guide covers epsilon-delta definitions, techniques for piecewise functions, and common pitfalls to avoid. Master 1.12 and solidify your understanding of continuity in calculus. (158 characters)

Understanding continuity is fundamental in calculus. This article delves into the methods for proving continuity of a function across a specified interval, specifically addressing the nuances of confirming continuity over an interval in the context of a calculus course (often denoted as 1.12 or a similar section number). We'll explore both theoretical and practical applications.

Understanding Continuity

Before diving into interval-specific proofs, let's refresh our understanding of continuity. A function f(x) is continuous at a point c if three conditions are met:

  1. f(c) is defined (the function exists at the point).
  2. The limit of f(x) as x approaches c exists (limx→c f(x) exists).
  3. The limit equals the function value at that point: limx→c f(x) = f(c).

A function is continuous on an interval if it's continuous at every point within that interval. This means we need to demonstrate the above conditions hold true for every point within the interval.

Proving Continuity Over an Interval: The Epsilon-Delta Approach

The most rigorous way to prove continuity is using the epsilon-delta definition. For a function f(x) to be continuous at a point c, for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - f(c)| < ε.

To prove continuity over an interval [a, b], you must show that this epsilon-delta condition holds for every c in the interval [a, b]. This is often challenging directly. Instead, we often employ other strategies.

Step-by-Step Guide to Epsilon-Delta Proofs (for specific points within the interval)

  1. Start with an arbitrary ε > 0: This represents the desired closeness of f(x) to f(c).
  2. Find an expression for |f(x) - f(c)|: Manipulate this expression to find a relationship with |x - c|.
  3. Choose a δ: Based on your manipulation in step 2, choose a δ that ensures |f(x) - f(c)| < ε whenever 0 < |x - c| < δ. This δ will often be expressed in terms of ε.
  4. Verify the implication: Show that if 0 < |x - c| < δ, then |f(x) - f(c)| < ε.

Example: Prove that f(x) = x² is continuous at x = 2.

  1. Let ε > 0 be arbitrary.
  2. |f(x) - f(2)| = |x² - 4| = |x - 2||x + 2|.
  3. Let's restrict x to be within a distance of 1 from 2. Then |x - 2| < 1 implies 1 < x < 3, so 3 < x + 2 < 5. Therefore, |x + 2| < 5. So, |f(x) - f(2)| < 5|x - 2|.
  4. Choose δ = min(1, ε/5). Then, if 0 < |x - 2| < δ, we have |f(x) - f(2)| < 5|x - 2| < 5(ε/5) = ε. Thus, f(x) = x² is continuous at x = 2. This process needs to be repeated for all points in the interval.

Proving Continuity for Piecewise Functions

Piecewise functions require careful consideration at the points where the definition changes. You must demonstrate continuity at each point of the interval, paying special attention to the transition points between different pieces of the function's definition.

Example: Consider a piecewise function:

f(x) = { x²,  x ≤ 2
        { 2x - 2, x > 2

To check continuity over an interval including x = 2, we need to specifically verify the three conditions for continuity at x = 2.

Common Pitfalls to Avoid

  • Incorrect assumptions: Don't assume continuity without proof.
  • Ignoring endpoints: Endpoints of closed intervals require special attention.
  • Overlooking piecewise functions: Carefully examine the behavior of piecewise functions at the transition points.

Conclusion

Proving continuity over an interval rigorously involves understanding the epsilon-delta definition and applying it appropriately. While directly applying the epsilon-delta definition for every point within an interval can be cumbersome, understanding the underlying principle is crucial. Mastering techniques for handling piecewise functions and avoiding common mistakes will solidify your grasp of continuity concepts, crucial for further progress in calculus. Remember to always focus on the fundamental definition of continuity and adjust your strategy based on the specific function and interval.

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