Chapter 3 and 4 of the book: How to Study as a Mathematics Major
Chapter 3 and 4 of the book “How to Study as a Mathematics Major” discusses defintions and theorems and gives multiple of examples that you come across while study mathematics courses such as calculus and introduction to proofs. When faced with a definition and working to understand the definition, you should consider examples that satisfy the definition and also examples that do not. This will help you with the abstract concepts in the definition as you build your understanding.
To approach this chapter, we present a definition that most of you have not seen before.
Definition: Let f be a real-valued function from the real numbers to the real numbers. Then the function f is called fine if it has a root (zero) at each integer. In other words, f is fine if for each integer n, f(n) = 0.
What are some of the key words in this definition that you have seen before in mathematics? State each key word and what it means.
Now, write down at least two examples of functions that are fine. Write down at least two examples of functions that are not fine. Describe how this helps you to understand the definition of a fine function. Also, describe issues that you still have in understanding this definition.
Most usually your examples are continuous functions. Write down two examples that is not continuous. Write down two examples that are not differentiable.
Describe your thoughts about a problem like this and how comfortable you would be in proving statements involving fine functions. What obstacles would you be faced with when proving such statements. For example, prove or disprove that the sum of two fine functions is also a fine function.
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