Limits and Continuity

1Introduction

This lecture explains that a limit describes the convergence of function values near a target point, whereas continuity requires that this limit agree with the function value at the target point.

An intuitive description of “approaching” is insufficient for rigorous analysis. The epsilon-delta formulation developed in the nineteenth century made it possible to analyze functions whose behavior contradicts geometric intuition, including continuous functions that are nowhere differentiable.

2Terms and Definitions

2.1Limit

Let $$f:D\to \mathbb{R}$$f:D\to\mathbb{R}, where $$D\subseteq \mathbb{R}$$D\subseteq\mathbb{R}, and let $$a$$a be an accumulation point of $$D$$D. The statement

means that, for every $$\epsilon >0$$\varepsilon>0, there is some $$\delta >0$$\delta>0 such that

Here $$\epsilon$$\varepsilon is the permitted output error and $$\delta$$\delta controls the input neighborhood. No matter how small an output tolerance is prescribed, a sufficiently small punctured neighborhood of $$a$$a forces the output error below that tolerance.

2.2Continuity

The function $$f$$f is continuous at $$a$$a precisely when the following three conditions hold.

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).

Keeping these conditions separate identifies the type of discontinuity when continuity fails.

2.3One-Sided Limits

The notation $$\underset{x\to a-}{\lim }f\left(x\right)=L$$\lim_{x\to a-}f(x)=L describes convergence as $$x$$x approaches $$a$$a through values smaller than $$a$$a. Similarly, $$\underset{x\to a+}{\lim }f\left(x\right)=L$$\lim_{x\to a+}f(x)=L describes approach through values larger than $$a$$a.

If $$a$$a is an accumulation point of $$D$$D from both the left and the right, a two-sided limit exists exactly when both one-sided limits exist and are equal. If the domain is an interval and $$a$$a is an endpoint, the relative limit on $$D$$D is the corresponding one-sided limit through points of the domain.

3Strategy

For elementary limit calculations, attempt direct substitution first, then algebraic cancellation after factorization, and finally the squeeze theorem when appropriate. To test continuity, verify the three defining conditions separately.

4Rigorous Explanation

4.11. A Minimal Epsilon-Delta Proof

To prove $$\underset{x\to 2}{\lim }(3x-1)=5$$\lim_{x\to2}(3x-1)=5, fix an arbitrary $$\epsilon >0$$\varepsilon>0 and construct $$\delta$$\delta so that $$|3x-1-5|<\epsilon$$|3x-1-5|<\varepsilon. Since $$|3x-1-5|=3|x-2|$$|3x-1-5|=3|x-2|, it is sufficient to require $$|x-2|<\epsilon /3$$|x-2|<\varepsilon/3. Set $$\delta =\epsilon /3$$\delta=\varepsilon/3. Then

The essential step is to derive a sufficient condition backward from the target inequality rather than guess $$\delta$$\delta.

4.22. Limit Calculation Techniques

Direct substitution: If $$f$$f is continuous at $$a$$a, then $$\underset{x\to a}{\lim }f\left(x\right)=f\left(a\right)$$\lim_{x\to a}f(x)=f(a). This applies to polynomials and to rational functions whose denominators are nonzero at $$a$$a.

Factorization: For $$\underset{x\to 1}{\lim }(x^2-1)/(x-1)$$\lim_{x\to1}(x^2-1)/(x-1), factor and cancel for $$x\ne 1$$x\ne1 to obtain $$x+1$$x+1; the limit is therefore $$2$$2.

Squeeze theorem: If $$g\left(x\right)\text{[PARSE ERROR: Undefined("Command("le")")]}f\left(x\right)\text{[PARSE ERROR: Undefined("Command("le")")]}h\left(x\right)$$g(x)\le f(x)\le h(x) near $$a$$a and both outer functions tend to $$L$$L, then $$f\left(x\right)$$f(x) also tends to $$L$$L. For example, $$-\left|x\right|\text{[PARSE ERROR: Undefined("Command("le")")]}x\sin (1/x)\text{[PARSE ERROR: Undefined("Command("le")")]}\left|x\right|$$-|x|\le x\sin(1/x)\le|x| implies $$\underset{x\to 0}{\lim }x\sin (1/x)=0$$\lim_{x\to0}x\sin(1/x)=0.

With angles measured in radians, the following basic limit holds:

4.33. Discontinuities Covered in This Lecture

The following classification concerns discontinuities at finite points, excluding those involving infinite limits.

4.44. Important Properties of Continuous Functions

Intermediate value theorem: If $$f$$f is continuous on $$[a,b]$$[a,b], then for every $$c$$c between $$f\left(a\right)$$f(a) and $$f\left(b\right)$$f(b) there is some $$\xi \in [a,b]$$\xi\in[a,b] with $$f\left(\xi \right)=c$$f(\xi)=c. If $$c$$c lies strictly between the endpoint values, then $$\xi \in (a,b)$$\xi\in(a,b).

Thus, a continuous function on an interval cannot omit an intermediate value. In particular, if $$f\left(0\right)<0<f\left(1\right)$$f(0)<0<f(1), a zero exists in $$[0,1]$$[0,1]; this result supports the bisection method.

Extreme value theorem: A continuous function on a closed bounded interval $$[a,b]$$[a,b] attains both its maximum and minimum, as a consequence of compactness.

4.55. Why Epsilon-Delta Is Necessary

Defining “approaches” only by saying that $$f\left(x\right)$$f(x) is close to $$L$$L whenever $$x$$x is close to $$a$$a is circular because “close” remains undefined. Epsilon-delta inequalities quantify both tolerances and reduce the assertion to inequalities that can be manipulated mathematically.

5Decision Criteria

• If direct substitution gives a zero denominator, test whether factorization permits cancellation.
• For a product of a quantity tending to zero and a bounded quantity, consider the squeeze theorem.
• If $$f\left(a\right)$$f(a) is undefined, distinguish existence of the limit from continuity.
• To establish the existence of a zero, consider the intermediate value theorem.

6Scope of the Results

The epsilon-delta definition of a limit does not itself require completeness of the real numbers. Existence theorems such as the intermediate value theorem do use completeness in their proofs, and analogous statements can fail over the rational numbers. Multivariable limits require control over all approaches to the target point, not merely agreement along a selected collection of directions.

7Final Form

8Summary

A limit formalizes convergence near a target point through inequalities, while continuity requires agreement between that limit and the function value. The intermediate value theorem guarantees that a continuous function on an interval omits no intermediate value.

9Exercises

10Related Material