Continuity at a point | Limits and continuity | AP Calculus AB | Khan Academy

TL;DR

Continuity in a function means being able to draw its graph at a point without picking up your pencil.

Transcript

• [Instructor] What we're going to do in this video is come up with a more rigorous definition for continuity. And the general idea of continuity, we've got an intuitive idea of the past, is that a function is continuous at a point, is if you can draw the graph of that function at that point without picking up your pencil. So what do we mean by tha... Read More

Key Insights

• 😥 The intuitive idea of continuity in a function is being able to draw its graph at a point without lifting the pencil.
• 👈 A function is continuous at a point if the two-sided limits as x approaches that point are equal to the value of the function at that point.
• 😥 If the two-sided limits approach different values or do not exist, then the function is not continuous at that point.

Questions & Answers

Q: What does it mean for a function to be continuous at a point?

If a function is continuous at a point, it means that you can draw its graph at that point without lifting your pencil. The two-sided limits of the function as x approaches that point are equal to the value of the function at that point.

Q: How can the two-sided limits determine continuity?

The two-sided limits of a function as x approaches a point show if the function approaches the same value from both the left and right sides. If the two-sided limits approach the same value, then the function is continuous at that point.

Q: What happens if the two-sided limits approach different values?

If the two-sided limits of a function approach different values as x approaches a point, then the function is not continuous at that point. It would require lifting the pencil to draw the graph of the function at that point.

Q: Can a function be continuous if the two-sided limits do not exist?

No, if the two-sided limits of a function do not even exist as x approaches a point, then the function is not continuous at that point. This means that the graph of the function would have a jump or a discontinuity at that point.

Summary & Key Takeaways

• Continuity in a function means that you can draw its graph at a specific point without lifting your pencil.

• A function is continuous at a point if the two-sided limits of the function as x approaches that point are equal to the value of the function at that point.

• If the two-sided limits approach different values or if the limit does not even exist, then the function is not continuous at that point.