Continuity of the Identity Function

When Each Input Is Its Own Output

For a constant function, every input produces the same output, so output differences vanish. The identity function behaves in the opposite way: it returns the input itself. Its output difference is therefore not zero in general, but it exactly matches the input difference. That equality makes the epsilon-delta proof especially direct.

As in the previous tutorial, the domain may be any subset of \(\mathbb{R}\). It need not contain an interval around a point, and it may be unbounded or even empty. The continuity condition tests only those inputs that belong to the domain.

If \(a,x\in E\), then \(i(x)=x\) and \(i(a)=a\). Consequently, \(|i(x)-i(a)|=|x-a|\). Thus, to make the output error smaller than \(\varepsilon\), it is enough to require the input to be within \(\varepsilon\) of \(a\). This suggests choosing \(\delta=\varepsilon\).

The Identity Preserves Distances

The key calculation applies not only when one input is a fixed continuity point, but to every pair of inputs in the domain. It is useful to state this as a result in its own right.

Proof. Fix \(x,y\in E\). By the definition of the identity function, \(i(x)=x\) and \(i(y)=y\). Substitution gives \[ |i(x)-i(y)|=|x-y|. \] This proves the equality for every pair \(x,y\in E\). If \(E\) is empty, there are no pairs to check, so the assertion is vacuously true. \(\square\)

A function that preserves distances is sometimes called an isometry. Here, the term describes the exact equality just proved; no assumptions about the shape of \(E\) are needed. In particular, the identity function does not stretch or shrink any distance, even when the domain has gaps or consists of isolated points.

Continuity at Each Domain Point

Proof. Fix \(a\in E\), and let \(\varepsilon>0\). Choose \(\delta=\varepsilon\), which is positive. Suppose \(x\in E\) and \(|x-a|<\delta\). Since \(i(x)=x\) and \(i(a)=a\), we have \[ |i(x)-i(a)|=|x-a|<\delta=\varepsilon. \] This is precisely the epsilon-delta condition for continuity at \(a\). Since \(a\) was arbitrary, \(i\) is continuous at every point of \(E\). If \(E\) is empty, there are no points at which to check continuity, and the statement holds vacuously. \(\square\)

The choice \(\delta=\varepsilon\) works because the input and output errors are equal. As in the epsilon-delta definition, \(\delta\) need not be the largest possible choice. For example, any positive \(\delta\leq\varepsilon\) also works: if \(|x-a|<\delta\), then \(|i(x)-i(a)|=|x-a|<\delta\leq\varepsilon\).

One Delta Works on the Whole Domain

Continuity at a point allows the choice of \(\delta\) to depend on that point. Uniform continuity requires a stronger arrangement: once \(\varepsilon\) is fixed, the same \(\delta\) must work for every pair of domain points. The distance-preserving calculation proves this stronger property immediately.

Proof. Fix \(\varepsilon>0\), and choose \(\delta=\varepsilon\). Let \(x,y\in E\) satisfy \(|x-y|<\delta\). By distance preservation, \[ |i(x)-i(y)|=|x-y|<\delta=\varepsilon. \] The choice of \(\delta\) depends only on \(\varepsilon\), not on \(x\) or \(y\). It therefore works for every pair in \(E\), which is the definition of uniform continuity. If \(E\) is empty, there are no pairs to check, so uniform continuity holds vacuously as well. \(\square\)

In fact, the same \(\delta=\varepsilon\) also gives continuity at every individual point. The difference is the scope of the claim: pointwise continuity fixes a center \(a\), whereas uniform continuity must handle all pairs \(x,y\) in the domain at once. For the identity, the exact distance equality makes the single choice work throughout.

Restricted Domains and Exact Delta Choices

A function’s domain matters because it determines which inputs must be tested. But restricting the identity function to a smaller set does not alter its rule at any point that remains in the domain. The distance equality continues to hold for every pair of points in that smaller set.

The finite example shows one way a continuity condition can be satisfied on a restricted domain: there may be few points to test near the chosen center. But the identity function does not rely on that feature. The same proof works whether a neighborhood contains one domain point, many domain points, or none other than its center.

Why the Distance Equality Matters

For the identity function, the central estimate is an equality: \[ |i(x)-i(a)|=|x-a|. \] There is no need to produce an indirect bound or first restrict \(x\) to a smaller region. The epsilon-delta choice follows directly from the desired output tolerance.

A common pitfall is to treat continuity as a statement about all real numbers near \(a\), even when the function’s domain is only \(E\). The correct implication tests \(x\in E\). For example, if \(E\) has gaps, values in those gaps are not inputs to the function and need not be considered. Another pitfall is to assume that pointwise continuity alone automatically gives uniform continuity for every function. That implication is not generally valid; here uniform continuity follows from the stronger, domain-wide distance calculation.

The identity function is also a useful reference point for later continuity arguments. Its output changes at exactly the same rate as its input: a given input distance is neither enlarged nor reduced. In the next tutorial, polynomial functions will be built from simpler functions, and their output differences will generally require more involved estimates than this exact equality.