4 min read•june 18, 2024
Now that we’ve explored different types of discontinuity, we can dive into what really is and how we can define it at a point. 🎯
Continuity means that a behaves smoothly and doesn't have sudden jumps or gaps in its graph. It's like drawing a line 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 important, especially in calculus and real analysis, to understand how functions change and interact.
On the AP exam, you’re going to have to be able to justify WHY something is continuous (or not continuous). Let's explore ways to figure out if a function is continuous!
A function f(x) is continuous at a specific point 'c' in its domain if the following three conditions are met:
1️⃣ f(c) is defined (i.e., there is a value of the function at c)
2️⃣ The as x approaches c exists
3️⃣ The value of the function at c (f(c)) is equal to the limit of the function as x approaches c. In other words, .
When working with FRQs, it can be helpful to bullet out these conditions and work through each of them. This helps you organize your thoughts and will make the grader’s work easier! 💫
To prove that a line is continuous at a specific point using a graph, you'll need to ensure that there are no jumps, gaps, or breaks in the graph at that point. The key is to visually demonstrate that the line flows smoothly without any interruptions. Here's how you can do it:
For example, take a look at these graphs:
Four graphs are displayed and the graph on the top left is the only one that is continuous.
Image courtesy of SFU.ca
In this example, only the top left graph is continuous. On all other graphs, there is a skip which makes the graph discontinuous at that point. Below the graphs are explanations as to why the graph is not continuous. These explanations are what we should aim for!
Is the function continuous at ? Justify your conclusion using the definition.
Explanation: To check continuity, we need to ensure three things:
Consider the functions and . Are both functions continuous at ? Justify your conclusions using the definition.
Explanation: For each function, apply the definition of continuity:
Perfect, both functions are continuous at x = 3. ✏️
Examine the functions and . Are both functions continuous at ? Justify your answer.
Explanation:
Therefore, function q(x) is continuous at x = 0, while p(x) is not continuous at x = 0.
Consider the context or problem you're dealing with. Look at the given information or scenario and check if it aligns with the hypotheses. This might involve looking at the values, functions, or objects involved.
If you find that the necessary conditions aren't met, it's really important to show why. You can do this by giving examples that prove the statement doesn't work. These examples are super useful when you want to show that a rule or idea isn't always true.
Good luck, you got this! 🍀