3 Step Continuity Test, Discontinuity, Piecewise Functions & Limits | Calculus

TL;DR

The three-step continuity test is used to determine if a function is continuous at a specific point.

Transcript

how can we prove that a function is continuous at a certain point how can we do so there's something called the three-step continuity test and the first step is that you have to show that the function is defined at some point a so f of a has to exist it has to equal a certain value the second step is to show that the limit as x approaches a of f of... Read More

Key Insights

• ⛔ The three-step continuity test requires the function to be defined at a certain point, the limit of the function to exist, and the limit from both sides to match the function value.
• 👍 Continuity can be proven by satisfying all three steps of the continuity test, while discontinuity can be identified if any step fails.
• ☺️ In the example, the function f(x) = √(x + 2) is continuous at x = 2, but discontinuous at x = 3.

Summary & Key Takeaways

• The three-step continuity test requires the function to be defined at the point in question, the limit of the function to exist as x approaches the point, and the limit from both sides to be equal to the function value.

• To apply the test, an example function is given: f(x) = √(x + 2) for x < 2, x^2 - 2 for 2 < x < 3, and 2x + 5 for x ≥ 3.

• Using the three-step continuity test, it is proven that the function is continuous at x = 2, as all three steps are satisfied.

• However, the function is found to be discontinuous at x = 3, as the limits from the left and right sides do not match.