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

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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 ...
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