How to Prove a Function is Continuous using Delta Epsilon

TL;DR

The video proves that the square root function is continuous everywhere it is defined using the definition of continuity.

Transcript

hey YouTube in this video we're going to prove that the square root function is continuous everywhere it's defined so this function is defined on the set bracket 0 infinity so first we'll prove it's continuous at 0 and then we'll give a proof to show it's continuous at every positive number therefore it will be continuous everywhere in its domain b... Read More

Key Insights

• 😫 The square root function is defined on the set [0, +∞].
• ❎ Continuity at 0 is proved by choosing Delta as epsilon squared and showing that the absolute value of the square root of X is less than epsilon.
• 👍 Continuity at positive numbers is proved by rationalizing the numerator and choosing Delta as epsilon times the square root of C.

Questions & Answers

Q: What is the definition of continuity of a function at a specific value?

Continuity of a function at a number C means that for any epsilon greater than zero, we can find a Delta such that for any X within a distance of Delta from C, the distance between f(X) and f(C) is less than epsilon.

Q: How is continuity at 0 proved for the square root function?

By choosing Delta as epsilon squared, we can show that if the absolute value of X is less than Delta, then the absolute value of the square root of X is less than epsilon.

Q: What is the approach to prove continuity at positive numbers for the square root function?

The approach involves rationalizing the numerator and choosing Delta as epsilon times the square root of C. This ensures that the distance between f(X) and f(C) is less than epsilon.

Q: What are the key steps in proving continuity at positive numbers for the square root function?

The key steps involve multiplying the expression by the square root of X plus the square root of C to rationalize the numerator, dropping the square root of X, and choosing Delta as epsilon times the square root of C.

Summary & Key Takeaways

• The video aims to prove that the square root function is continuous at 0 and every positive number in its domain.

• The proof for continuity at 0 involves choosing Delta as epsilon squared and showing that the distance between f(X) and f(0) is less than epsilon.

• The proof for continuity at positive numbers involves rationalizing the numerator and choosing Delta as epsilon times the square root of C.