# sol_hom1 - Solutions to Homework 1 17.3 Assume that log e...

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

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.

Unformatted text preview: Solutions to Homework 1 January 16, 2009 17.3: Assume that log e , cos , sin , and x p are all continuous functions. a: Continuity through Theorem 17.4(i) and 17.5 b: Continuity through Theorem 17.5 on cos 6 and sin 2 , then 17.4(i) and finaly 17.5 again 17.4: Prove that the function ( x ) 1 / 2 is continuous on [0 , ∞ ) Proof: case 1: x = 0: for any ǫ > , ∃ δ < ǫ 2 s.t. for all | x − | < δ, | ( x ) 1 / 2 − (0) 1 / 2 | = ( x ) 1 / 2 < ( δ ) 1 / 2 = ǫ case 2: x ∈ (0 , ∞ ): for any ǫ > ∃ δ ≤ ǫ ( x ) 1 / 2 such that ∀| x − x | < δ, | ( x ) 1 / 2 − ( x ) 1 / 2 | = | ( x ) 1 / 2 − ( x ) 1 / 2 | 1 | ( x ) 1 / 2 + ( x ) 1 / 2 | | ( x ) 1 / 2 + ( x ) 1 / 2 | = | x − x | ( x ) 1 / 2 + ( x ) 1 / 2 ≤ x − x ( x ) 1 / 2 < δ ( x ) 1 / 2 ≤ ǫ ( x ) 1 / 2 ( x ) 1 / 2 = ǫ 17.9a Prove that f ( x ) = x 2 is continuous at x = 2 using ǫ − δ property: Note that if | x − 2 | < 1 then by the triangle inequality | x | < 1 + 2 = 3 and using the triangle eniquality again we get | x + 2 | ≤ | x...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

sol_hom1 - Solutions to Homework 1 17.3 Assume that log e...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online