Functions, Domains, and Interpreting Graphs

1Introduction

Before introducing limits and differentiation, this lecture clarifies which inputs belong to the domain and at which point a limit is being considered.

A function is a rule that assigns exactly one output to each input in its domain. A formula alone, however, does not determine a function. One must also specify its domain, the set of permitted inputs.

2Intuitive Interpretation from a Graph

On a graph, a limit describes the behavior of function values near a point, rather than the value that the function takes at that point. Even at a point excluded from the domain but approachable from both sides, the limit can exist when the function values from the left and right converge to the same value.

When the domain is an interval, each endpoint can be approached from only one direction within the domain. At a right endpoint, examine behavior from the left; at a left endpoint, examine behavior from the right. This is why one-sided limits are necessary.

3Precise Formulation

Let the domain be $$D$$D. To consider the limit at $$a$$a, the necessary condition is not that $$a$$a belong to $$D$$D, but that $$a$$a be an accumulation point of $$D$$D. Equivalently, for every $$\delta >0$$\delta>0, there must be some $$x\in D$$x\in D satisfying $$0<|x-a|<\delta$$0<|x-a|<\delta.

To verify that $$f$$f is continuous at $$a$$a, check the following three conditions separately.

1. $$f\left(a\right)$$f(a) is defined.
2. $$\underset{x\to a}{\lim }f\left(x\right)$$\lim_{x\to a}f(x) exists.
3. $$\underset{x\to a}{\lim }f\left(x\right)=f\left(a\right)$$\lim_{x\to a}f(x)=f(a).

4Example

Let $$D=\mathbb{R}\setminus \left\{1\right\}$$D=\mathbb{R}\setminus\{1\} and define $$f:D\to \mathbb{R}$$f:D\to\mathbb{R} by $$f\left(x\right)=(x^2-1)/(x-1)$$f(x)=(x^2-1)/(x-1). Since the denominator is zero at $$x=1$$x=1, the number $$1$$1 is excluded from the domain. For $$x\ne 1$$x\ne1, however, $$f\left(x\right)=x+1$$f(x)=x+1. Thus, around the hole, the function has the same behavior as the line $$y=x+1$$y=x+1.

This example demonstrates the necessity of distinguishing a function value, the domain, and a limit.

5Exercises

6Related Material