Continuity and discontinuity

Textbook reading

The problem that opens the lesson

Choose $k$ so that $f(x)=\frac{x^2-4}{x-2}$ for $x\ne 2$ and $f(2)=k$ is continuous.

Solution

Begin by identifying the mathematical object and the information that fixes it. Check the function value, left limit, and right limit separately, then classify the first failed condition and determine whether a redefinition can repair it. The relevant conditions are not optional bookkeeping: Continuity depends on the declared domain. A function can be continuous everywhere in its domain while having excluded endpoints or asymptotes outside that domain. Following that structure gives $k=4$ .

Why this works

At domain endpoints, one-sided continuity is the relevant concept. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

What this lesson is really about

A function is continuous at an interior point $c$ when $f(c)$ exists, the two-sided limit exists, and the limit equals $f(c)$ .

Removable discontinuities have a common approached value but a missing or incorrect point. Jump discontinuities have unequal one-sided limits. Infinite discontinuities involve unbounded behavior.

The point is not merely to reproduce a formula. A learner should be able to identify the quantities or geometric objects involved, explain why the relationship has its stated form, and recognize when the same idea appears in a graph, table, diagram, or model.

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Why the relationship works

At domain endpoints, one-sided continuity is the relevant concept.

Textbook reading

A reliable way to work

Check the function value, left limit, and right limit separately, then classify the first failed condition and determine whether a redefinition can repair it.

Continuity depends on the declared domain. A function can be continuous everywhere in its domain while having excluded endpoints or asymptotes outside that domain.

After the symbolic work is complete, check the result. Depending on the lesson, this may mean substituting into an original equation, comparing coordinates, examining a graph, checking units, testing an interval, or confirming that every branch of a periodic solution has been included.

Textbook reading

What commonly goes wrong

A common error is calling a graph discontinuous merely because it has a sharp corner; corners can be continuous.

The repair is to return to the definition and identify the first step where the invalid solution stops describing the original mathematical object. Later algebra cannot rescue a first step that changed the domain, orientation, branch, or meaning of the problem.

Textbook reading

Worked examples

Worked example 1

Choose $k$ so that $f(x)=\frac{x^2-4}{x-2}$ for $x\ne 2$ and $f(2)=k$ is continuous.

Solution

Begin by identifying the mathematical object and the information that fixes it. Check the function value, left limit, and right limit separately, then classify the first failed condition and determine whether a redefinition can repair it. The relevant conditions are not optional bookkeeping: Continuity depends on the declared domain. A function can be continuous everywhere in its domain while having excluded endpoints or asymptotes outside that domain. Following that structure gives $k=4$ .

Why this works

At domain endpoints, one-sided continuity is the relevant concept. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Transfer example

Problem

Classify a piecewise jump.

Worked development

Check the function value, left limit, and right limit separately, then classify the first failed condition and determine whether a redefinition can repair it. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. Removable discontinuities have a common approached value but a missing or incorrect point. Jump discontinuities have unequal one-sided limits. Infinite discontinuities involve unbounded behavior. Then apply the conditions explicitly: Continuity depends on the declared domain. A function can be continuous everywhere in its domain while having excluded endpoints or asymptotes outside that domain. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Continuity supports existence arguments, model reliability, and later theorems of Calculus.

Reasoning example

Problem

Classify a rational vertical asymptote.

Worked development

Check the function value, left limit, and right limit separately, then classify the first failed condition and determine whether a redefinition can repair it. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. Removable discontinuities have a common approached value but a missing or incorrect point. Jump discontinuities have unequal one-sided limits. Infinite discontinuities involve unbounded behavior. Then apply the conditions explicitly: Continuity depends on the declared domain. A function can be continuous everywhere in its domain while having excluded endpoints or asymptotes outside that domain. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Continuity supports existence arguments, model reliability, and later theorems of Calculus.

Worked example 4: quick check

What three conditions are needed for continuity at an interior point $c$ ?

Solution

Begin by identifying the mathematical object and the information that fixes it. Check the function value, left limit, and right limit separately, then classify the first failed condition and determine whether a redefinition can repair it. The relevant conditions are not optional bookkeeping: Continuity depends on the declared domain. A function can be continuous everywhere in its domain while having excluded endpoints or asymptotes outside that domain. Following that structure gives $f(c)$ exists, the limit exists, and the limit equals $f(c)$ .

Why this works

At domain endpoints, one-sided continuity is the relevant concept. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

Application and interpretation

Continuity supports existence arguments, model reliability, and later theorems of Calculus.

A contextual answer must include units, a meaningful domain, and the assumptions that make the model plausible. An exact mathematical relationship should not be diluted into a decimal unless a measurement or comparison requires it.

Practice

Ten concrete questions

Practice 101

What three conditions are needed for continuity at an interior point $c$ ?

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