Introduction to Real Analysis Dr. Weihu Hong Clayton State University 11/11/2008

Uniform Continuity Definition Let E be a subset of R and f a real-valued function with domain E, that is, f : E  R. The function f is uniformly continuous on E if Remark. The choice of δ depends only on ε. Question: how would you define that f is not uniformly continuous on E?

Examples for discussion

Lipschitz Functions Function f : E  R satisfies a Lipschitz condition on E if Functions satisfy the Lipschitz condition are called Lipschitz functions.

Theorem on Lipschitz Functions Theorem Suppose f : E  R is a Lipschitz function on E, then it is uniformly continuous on E. Example.

Uniform Continuity Theorem Theorem If K is a compact subset of R and f: K  R is continuous on K, then f is uniformly continuous on K. Corollary A continuous real-valued function on a closed and bounded interval [a, b] is uniformly continuous.

The Derivative Definition Let I be an interval and let f : I  R. For fixed p є I, the derivative of f at p, denoted f’(p), is defined to be provided the limit exists. If f’(p) is definded at a point p є I, we say that f is differentiable at p. If the derivative f’ is defined at every point of a set E, we say that f is differentiable on E.

The Right Derivative Definition Let I be an interval and let f : I  R. For fixed point p є I (p is not the right end point of I), the right derivative of f at p, denoted provided the limit exists.

The Left Derivative Definition Let I be an interval and let f : I  R. For fixed point p є I (p is not the left end point of I), the left derivative of f at p, denoted provided the limit exists.

The Derivative Remark. Let I be an interval and let f : I  R. For fixed point p є int(I), the derivative of f at p exists if and only if Theorem If I is an interval and f:I  R is differentiable at p є I, then f is continuous at p. Remark: The converse of the theorem is not true.

Derivatives of Sums, Products, and Quotients Theorem Suppose f,g are real-valued functions defined on an interval I. If f and g are differentiable at x є Int(I), then f+g, fg, and f/g (if g(x)≠0) are differentiable at x and

The Chain Rule Theorem Suppose f is a real-valued function defined on an interval I and g is a real-valued function defined on some interval J such that Range f is a subset of J. If f is differentiable at x є Int(I) and g is differentiable at f(x), then g ◦ f is differentiable at x and