Theorem Selection for Continuity Problems

Choose the Tool by Matching the Claim

Continuity problems often become easier once the right question is identified. A pointwise question concerns behavior near one specified input. A uniform continuity question asks for one radius that works across the whole domain. Other problems give structural hypotheses, such as compactness of the domain, a Lipschitz estimate, or a denominator that never approaches zero. Each hypothesis points toward a different theorem.

The aim here is to select a theorem before beginning a proof. This does not replace checking hypotheses: it makes that check the first step. Ask what the conclusion actually requires, identify the strongest relevant information in the assumptions, and verify that the theorem’s domain and range match the problem. Earlier in this course, the Sequential Criterion for Uniform Continuity, the Heine-Cantor Theorem, and Lipschitz Implies Uniform Continuity were established. We will use those results by name, and prove a new criterion for reciprocals and quotients.

The table is a guide, not a collection of interchangeable methods. For instance, pointwise continuity at every point does not itself provide a single radius for the entire domain. If compactness is available, Heine-Cantor can supply that missing uniformity; if the domain is not compact, another argument is needed.

Use Compactness When Its Hypotheses Fit

The Heine-Cantor Theorem is often the shortest route from continuity to uniform continuity, but only when the domain is compact. In \(\mathbb{R}\), a closed bounded interval is compact. Thus, for a function continuous at every point of such an interval, one can invoke the theorem directly. There is no need to construct an explicit radius for each \(\varepsilon\), unless the problem asks for one.

A common selection error is to notice continuity and stop without checking the domain. Heine-Cantor does not say that every continuous function on every domain is uniformly continuous. For an unbounded domain, or for a bounded domain that is not compact, this theorem cannot be applied as stated. The failure of a hypothesis does not prove that the conclusion is false; it only means that this particular theorem has not settled the question.

Use Sequences to Test a Global Claim

For a claim of nonuniform continuity, the Sequential Criterion for Uniform Continuity is particularly effective. It says that a function is not uniformly continuous if and only if there are sequences \(x_n,y_n\) in its domain such that \(|x_n-y_n|\to0\), while the output distances \(|f(x_n)-f(y_n)|\) do not tend to zero. The key is to make the input points approach one another while keeping a definite output separation.

A sequence test must address both parts of the criterion. Showing only that the input distances tend to zero does not disprove uniform continuity; the output distances must fail to tend to zero as well. Conversely, if the problem asks only about continuity at one fixed point, a pair of sequences moving farther and farther along the domain is not the relevant test.

A Uniform Lower Bound Controls Reciprocals

Reciprocals suggest a direct estimate. Their difference can be written using the difference of the original function values, but the denominator in that estimate must not become arbitrarily small. A uniform positive lower bound on the absolute value supplies exactly the needed control.

Proof. Let \(\varepsilon>0\). By uniform continuity of \(f\), there exists \(\delta>0\) such that for all \(x,y\in E\),

For any such \(x,y\), the lower bound ensures that neither \(f(x)\) nor \(f(y)\) is zero. Hence

The chosen \(\delta\) is independent of \(x\) and \(y\), so \(r\) is uniformly continuous on \(E\). \(\square\)

The size of the lower bound matters: the proof uses \(c^2\) because the product \(|f(x)f(y)|\) is at least \(c^2\). Merely knowing that \(f(x)\neq0\) at every individual point is not enough to use this argument; the values could approach zero across the domain.

When a Quotient Needs One More Hypothesis

For a quotient \(f/g\), the reciprocal theorem controls \(1/g\) if \(g\) is uniformly continuous and bounded away from zero. However, a quotient also contains the numerator. A useful sufficient condition is that the numerator be bounded as well as uniformly continuous. The following estimate makes both roles explicit.

Proof. If \(E\) is empty, the assertion holds vacuously. Otherwise, let \(\varepsilon>0\). Uniform continuity of \(f\) gives \(\delta_f>0\) such that \(|x-y|<\delta_f\) implies \(|f(x)-f(y)|<c\varepsilon/2\). Uniform continuity of \(g\) gives \(\delta_g>0\) such that \(|x-y|<\delta_g\) implies \(|g(x)-g(y)|<c^2\varepsilon/(2(M+1))\). Put \(\delta=\min\{\delta_f,\delta_g\}\), which is positive.

For \(x,y\in E\) with \(|x-y|<\delta\), use the identity

Since \(|g(x)|\geq c\), \(|f(y)|\leq M\), and \(|g(x)g(y)|\geq c^2\), the triangle inequality gives

The same \(\delta\) works for all \(x,y\in E\). Thus \(h\) is uniformly continuous on \(E\). \(\square\)

A Practical Theorem-Selection Checklist

Before writing a proof, classify the requested conclusion and inspect the assumptions. If the question concerns one point, use a pointwise criterion. If it concerns every pair in a domain, look for a uniform estimate, compactness, or a sequential test. If the function is built from other functions, check whether the earlier composition or sum theorem applies and whether all domains and ranges align. For a reciprocal or quotient, check nonvanishing quantitatively, not just pointwise.

A failed theorem match should lead to a more precise question: which hypothesis is missing, and can it be supplied by the domain or a direct estimate? For example, a compactness theorem cannot be used on the entire real line, and the reciprocal theorem cannot be used without a positive lower bound. These observations do not decide the problem by themselves, but they prevent an invalid shortcut and point toward the estimate that would be needed.