Delta Epsilon Proof that f(x) = sin(x) is a Continuous Function using the Definition of Continuity

TL;DR

This video provides a proof that the sine function is continuous at every real number, using trig identities and the Delta Epsilon definition of continuity.

Transcript

hey YouTube in this video we're going to prove that the sine function is continuous at every single real number recall that a function is continuous at x equals C so let me just say f is continuous at x equals C if for every epsilon greater than zero we can find some number Delta greater than 0 such that for every real number X so for all X and R w... Read More

Key Insights

• ❓ The proof relies on the Delta Epsilon definition of continuity.
• ❓ Choosing Delta as Epsilon simplifies the proof.
• 👨‍💼 Trig identities are used to transform the expression involving sine X and sine C.
• 🟰 The absolute value of cosine is less than or equal to 1, and the absolute value of sine X is less than or equal to the absolute value of X.

Questions & Answers

Q: What is the Delta Epsilon definition of continuity?

The Delta Epsilon definition states that a function is continuous at a given point if, for any positive error margin (Epsilon), there exists a positive variation in the input (Delta) that ensures the function's output remains within the error margin.

Q: How is Delta chosen in this proof?

In this proof, Delta is chosen to be equal to Epsilon. This choice simplifies the calculation and does not require extensive intuition.

Q: What trig identity is used in the proof?

The trig identity used is sine of A minus sine of B equals 2 times cosine of A plus B over 2 times sine of A minus B.

Q: How are trig identities applied in the proof?

The trig identities are used to rewrite the equation involving the absolute values of sine X and sine C in terms of cosine and sine. This allows for the comparison with the error margin (Delta) and simplifies the expression.

Summary & Key Takeaways

• The video aims to prove that the sine function is continuous at every real number.

• The proof is based on the Delta Epsilon definition of continuity and utilizes trig identities.

• The proof is kept simple, with the choice of Delta as Epsilon, and the use of trig identities to simplify the expression.

• By showing that the absolute value of the difference between sine X and sine C is less than Delta (which is equal to Epsilon), the proof is completed.