Definitions 10/26/17

The Intermediate Value Theorem:

In the book, we have the formal definition stating: Let f : [a, b] → R be continuous. If L is a real number satisfying f(a) < L < f(b) or f(a) > L > f(b), then there exists a point c ∈ (a, b) where f(c) = L.

To prove this theorem, we mainly have three possible approaches:

  1. Using the axiom of completeness.
  2. Using the nested interval property.

Details are required in the written homework.

Then, we moved on to chapter 5 where we discussed differentiation. The definition of differentiability is stated as:

Let g : A → R be a function defined on an interval A. Given c ∈ A, the derivative of g at c is defined by g'(c) = \lim_{x\rightarrow c}\frac{g(x)-g(c)}{x-c} provided this limit exists. In this case we say g is differentiable at c. If g' exists for all points c ∈ A, we say that g is differentiable on A.

Then, Theorem 5.2.3 states that If g : A → R is differentiable at a point c ∈ A, then g is continuous at c as well.

To prove this, Algebraic Limit Theorem for functional limits allows us to write \lim_{x\rightarrow c}(g(x)-g(c))=\lim_{x\rightarrow c}(\frac{g(x) - g(c)}{x-c})(x-c) = 0. Therefore, \lim_{x\rightarrow c}g(x) = g(c).

Algebraic Differentiability Theorem:

Let f and g be functions defined on an interval A, and assume both are differentiable at some point c ∈ A. Then:

  1. (f+g)'(c) = f'(c)+g'(c)
  2. (kf)'(c) = kf'(c) \forall k\in\mathbb{R}
  3. (fg)'(c) = f'(c)g(c) + f(c)g'(c)
  4. (f/g)'(c) = \frac{g(c)f'(c)-f(c)g'(c)}{[g(c)]^2}\text{, provided that }g(c)\neq0

 

Reference:

Abbott, Stephen. “Chapter 4 & Chapter 5.” Understanding Analysis, Springer, 2015.

 

Posted on by Yunwen Wan
Categories: Definitions

One Response

  1. Jeremy LeCrone says:
    October 31, 2017 at 9:13 pm

    I appreciate the effort that you put into this blog post, but it is still not meeting the expectations of the assignment. I expect that daily definitions blogs highlight a particular concept we have learned, and explore the details and implications of the definition. You’ve provided a brief synopsis of what happened in class, but this is not a “Daily Definitions” blog.

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