Proof that every Differentiable Function is Continuous

TL;DR

Differentiable functions at a point are also continuous due to the limit existing at that point.

Transcript

in this video we're going to prove that every differentiable function is continuous the actual statement we will prove is that if we have a function f which is differentiable at xal C then it is continuous at xal C as well so proof so we'll start by assuming that we have uh differentiability at C so suppose f is differentiable at xal C all right th... Read More

Key Insights

• 😥 Differentiability at a point indicates continuity at that point.
• 👍 Utilizing limits can simplify proving continuity in functions.
• 🪜 Adding zero is a common technique in calculus proofs for simplification.

Questions & Answers

Q: What is the correlation between differentiability and continuity in functions?

Differentiability at a point implies the existence of a limit, proving that the function is continuous at that point.

Q: How can incorporating limits help in proving continuity?

Incorporating limits, such as showing the limit of the difference to be zero, aids in proving continuity at a given point.

Q: Why is adding zero a common technique in calculus proofs?

Adding zero helps simplify calculations and proofs, especially when working with limits and continuous functions.

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

A function is continuous at a point if the limit as X approaches that point of f(x) equals F(C), showing a smooth transition.

Summary & Key Takeaways

• Proof that a function differentiable at a point is also continuous at that point.

• Differentiability implies the existence of a limit, allowing for continuity.

• By showing the limit as X approaches C of f(x) equals F(C), continuity at C is proven.