{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# 131a9 - Problems graded 29.3(15 points 29.13(10 points...

This preview shows pages 1–2. Sign up to view the full content.

Problems graded: 29.3 (15 points), 29.13 (10 points), 32.2 (10 points); the other problems were graded based on completion and worth five points each (except that I counted 28.8, which was longer, as two problems). To convert from 65 points to 100, scores were multiplied by 1.5; then we gave everyone 2.5 points, rounding all fractions up at the end. Final homework grade: Students have received nine scores, each out of 100. As per the syllabus, we dropped the two lowest scores for each students and converted the remaining sum to a percentage out of 100, rounding any fractional part up (623/700 = 89 percent but 624/700 = 89.14 percent gets bumped up to 90 percent). 28.8. (Note: I did this question in section on November 18.) (a) Suppose ǫ > 0. If | x 0 | < ǫ then f ( x ) is equal to either x 2 [0 , ǫ ) or 0; in any event, | f ( x ) f (0) | < ǫ implying continuity at zero. (b) x rational nonzero: Let ǫ = x 2 and suppose δ > 0. There exists an irrational number q in ( x δ, x + δ ) (see Exercise 4.12 from HW 2). However, while | x q | < δ , | f ( x ) f ( q ) | = ǫ ; δ can be made arbitrarily small so f is NOT continuous at x . x irrational: Let ǫ = . 1 x 2 and suppose δ > 0 (we may further suppose that δ < . 5 | x | ). There exists a rational number q in ( x δ, x + δ ); by the triangle inequality, | q | > . 5 | x | so f ( q ) = . 25 x 2 . Therefore, | f ( x ) f ( q ) | = . 25 x 2 > ǫ ; as δ can be made arbitrarily small, f is not continuous at x . As all nonzero points are either nonzero rational or irrational, f is discon- tinuous for all nonzero x . (c) If x = 0 and h negationslash = 0, f ( x + h ) f ( x ) h = f ( h ) h , which is equal to h if h is rational and 0 otherwise. In any event, the expression clearly goes to zero as h goes to zero (as it is bounded above in magnitude by | h | ) so f is differentiable at x = 0 with derivative 0.

This preview has intentionally blurred sections. Sign up to view the full version.

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

{[ snackBarMessage ]}