Lecture 18: Weierstrass's Example of a Continuous and Nowhere Differentiable Function

TL;DR

Differentiability and continuity are closely connected, but not all continuous functions are differentiable.

Transcript

[SQUEAKING] [RUSTLING] [CLICKING] CASEY RODRIQUEZ: So we're going to continue with our discussion of the derivative. So now, let me recall the definition we introduced at the end of last time of the derivative. So let I be an interval, meaning it could be open, closed, it could go out to plus infinity, it could go out to minus infinity. But you kno... Read More

Key Insights

• ☠️ The derivative is defined as the limit of the difference quotient, which measures the rate of change of a function.
• 😥 A function is differentiable at a point if and only if it is continuous at that point.
• ❓ The Weierstrass function is an example of a function that is continuous but not differentiable.
• 😥 The proof of the Weierstrass function's nowhere differentiability involves constructing a sequence of points and showing that the difference quotient is unbounded.

Questions & Answers

Q: What is the definition of the derivative?

The derivative of a function is defined as the limit of the difference quotient, which is the difference between the function values divided by the difference in the corresponding input values.

Q: What is the connection between differentiability and continuity?

If a function is differentiable at a point, it must also be continuous at that point. However, a continuous function may not be differentiable at all points.

Q: Can you provide an example of a function that is continuous but not differentiable?

Yes, the video discusses the Weierstrass function, which is continuous everywhere but nowhere differentiable. It is constructed using a series of cosine functions.

Q: How does the proof of the Weierstrass function's nowhere differentiability work?

The proof involves finding a sequence of points that converges to a given point and shows that the difference quotient of the function at these points is unbounded. This implies that the function is not differentiable at the given point.

Summary & Key Takeaways

• The video discusses the definition of the derivative and its connection to differentiability.

• It explains the theorem that states that a differentiable function is also continuous.

• The video introduces the example of a function that is continuous but not differentiable, known as the Weierstrass function.

• It provides a proof of the Weierstrass function's nowhere differentiability using estimates and series.