If function u is continuous at x, then _u_0 as _x_0 | AP Calculus AB | Khan Academy

TL;DR

If a function is continuous at a point, then as the change in x approaches zero, the change in the function also approaches zero.

Transcript

• [Voiceover] The result that I hope to show you, or you give you an intuition for, in this video is something that we will use in the proof of the chain rule, or in a proof of the chain rule actually. We may do more than proof of the chain rule. But the result we're gonna look at is if we have some function u which is a function of x, and we know ... Read More

Key Insights

• 💱 Continuity in a function at a point implies that the change in the function approaches zero as the change in x approaches zero.
• ☺️ The limit of u(x) - u(c) as x approaches c is an algebraic manipulation of the definition of continuity.
• 💱 The concept of the change in a function becoming smaller as the change in x becomes smaller is intuitive for continuous functions.

Questions & Answers

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

A function is continuous at a point if the limit of the function as x approaches that point is equal to the value of the function at that point, indicating no point or jump discontinuity.

Q: How can the limit of the function as x approaches c be rewritten to signify the change in the function?

The limit of u(x) - u(c) as x approaches c is equal to zero, signifying that as the change in x approaches zero, the change in the function also approaches zero.

Q: Why is the concept of continuity and change in functions important?

Understanding the intuition behind continuity and change in functions is crucial in proving mathematical concepts like the chain rule.

Q: What happens to the change in the function as the change in x approaches zero?

As the change in x approaches zero, the change in the function also approaches zero, assuming the function is continuous.

Summary & Key Takeaways

• Continuous functions have no point or jump discontinuities and the limit of the function as x approaches a point is equal to the value of the function at that point.

• The limit of the function as x approaches c can be written as the limit of the difference between the function at x and the function at c, which approaches zero as the change in x approaches zero.

• The concept of continuity and change in functions is important in understanding the chain rule.