# Category Archives: Advanced Calculus

## Local Existence and Uniqueness of IVP

In this note, we prove the local existence and uniqueness of initial value problem of  ODEs     where satisfies Lipschitz condition. 1. Function spaces To discuss our main topic, we need to extend some concepts that we already studied. In linear algebra, we studied vector spaces, which satisfies nine axioms. Such concepts give our… Read More »

## Space-filling curve

Theorem. There exists a continuous curve in that passes through every point of the unit square . Proof. Define by     Extend to all of by making periodic with period . Define     By the Weierstrass M-test, both and converges uniformly on . Moreover, it is continuous on . Now define and let… Read More »

## Weierstrass approximation theorem

In this note, we prove the Weierstrass approximation theorem. 1. Bernstein’s approximation Theorem. Let be continuous and let be given. Then there exists a polynomial such that . We can construct a polynomial explicitly. For each , we define     We call this polynomial as Bernstein polynomials. The polynomial can be interpreted in terms of… Read More »

## Counterexamples in differentiation

Exercise. Prove or disprove. If it is not true, find a counterexample. and , then . If is differentiable at , or are differentaible at . If and are infinitely differentiable on and on , then or . If is differentiable at and is differentiable at , then is differentiable at . If is differentiable… Read More »

## Cauchy Regular and continuity

Definition. Let be metric spaces and a function. We say is Cauchy regular provided that any Cauchy sequence in , is a Cauchy sequence in . Note that if is uniformly continuous on , then it is Cauchy regular. Indeed, let be a Cauchy sequence in and let be given. Since is uniformly continuous, there… Read More »