HowToDeltaEpsilon - How to Do ε, δ Proofs for Continuous...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: How to Do ε, δ Proofs for Continuous Functions Using the Difference Quotient In order to prove f is continuous at x = a we need to find a δ > 0 for each ε > 0 such that |f (x) − f (a) | < ε for all x satisfying |x − a| < δ . Step 1: Check to make sure the function is actually continuous at x = a. Step 2: Pick a value δmax such that the function does not have an infinite discontinuity in the interval (a − δmax , a + δmax ). Setting δmax = 1 usually works and makes the math simpler. Step 3: Form the difference quotient (this is the slope of the secant line through the points (a, f (a)) and (x, f (x)).) Step 4: Find a number m that is larger than the maximum value the absolute value of the difference quotient achieves for x in the interval (a − δmax , a + δmax ). Step 5: The answer is given by δ= δmax · δmax ε m if ε < m if ε ≥ m Example 1: Prove f (x) = x3 − 2x2 + 5 is continuous at x = 3. . Example 2: Prove f (x) = 1 is continuous at x = 5 . 2 x−2 . Example 3: Why does this method fail if used to prove f (x) = continuous at x = 0? Prove this using another approach. |x| is ...
View Full Document

This note was uploaded on 02/05/2011 for the course MATH 1A taught by Professor Wilkening during the Spring '08 term at University of California, Berkeley.

Ask a homework question - tutors are online