Defining Continuity at a Point
Abigail Young
7 min read
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Study Guide Overview
This study guide covers continuity of functions at a point. It defines continuity using a three-part test: f(c) is defined, the limit of f(x) as x approaches c exists, and the limit equals f(c). It explains how to determine continuity from graphs and provides practice questions involving algebraic functions and piecewise functions. The guide also includes multiple-choice and free-response practice problems with solutions and scoring guidelines.
# 1.11 Defining Continuity at a Point
Alright, let's nail down what it means for a function to be continuous. We've seen different types of discontinuities, and now it's time to define continuity at a point. 🎯
#What is Continuity?
Continuity means a function behaves smoothly, without sudden jumps or gaps. Think of it like drawing a curve without lifting your pen. To check if a function is continuous, we make sure it's not broken or full of holes, and it behaves predictably as we get closer to specific points. This concept is crucial in calculus for understanding how functions change and interact.
On the AP exam, you'll need to justify WHY a function is continuous (or not). Let's explore how to determine if a function is continuous! This is a key concept for both multiple-choice and free-response questions.
#Defining Continuity
A function f(x) is continuous at a point 'c' in its domain if these three conditions are met:
1️⃣ f(c) is defined (i.e., there's a function value at c).
2️⃣ The limit of the function as x approaches c exists. This means the left-hand limit and right-hand limit must be equal.
3️⃣ The function's value at c (f(c)) equals the limit as x approaches c. In other words, .
When tackling FRQs, bullet out these conditions and address each one. This keeps your thoughts organized and makes it easier for the grader to follow your reasoning! 📝
#Defining Continuity With A Graph
To show that a line is continuous at a specific point using a graph, you need to ensure there are no jumps, gaps, or breaks at that point. Visually...
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