Real Analysis 30 | Continuous Images of Compact Sets are Compact [dark version]

TL;DR

If I ⊂ ℝ is compact and f: I → ℝ is continuous, then the image f[I] is compact.

Briefing Cornell Notes

Briefing

A continuous function sends compact sets to compact sets—a property that guarantees the function attains both a maximum and a minimum on any compact domain. In practical terms, if a set of inputs is “well-behaved” in the sense of compactness, then the collection of outputs produced by a continuous map remains equally well-behaved. This matters because it turns abstract topological control (compactness) into concrete optimization results (existence of extrema).

The result is framed for subsets of the real numbers: start with a compact set I ⊂ ℝ and a function f: I → ℝ that is continuous at every point. The claim is that the image f[I] is compact. Compactness on the real line is tied to the Heine–Borel characterization: a set is compact exactly when it is closed and bounded. That connection helps explain why the image inherits “no escape” behavior—its outputs can’t run off to infinity and can’t accumulate at points outside the set.

The discussion then connects compactness to the supremum and infimum of the image. For a bounded set, the supremum and infimum exist; for a closed set, those limiting values actually belong to the set. Since f[I] is both bounded and closed, the supremum and infimum of f[I] become genuine maximum and minimum values. Geometrically, if I is a closed interval, the graph of f over I has a highest and lowest y-value, giving numbers x+ and x− in I such that f(x+) is the maximum and f(x−) is the minimum. The transcript also notes that this “attains extrema” conclusion can fail for non-continuous functions, where maxima and minima may exist only as limits rather than achieved values.

A proof is built directly from the sequential definition of compactness. Take any sequence (y_n) in the image f[I]. Because each y_n comes from applying f to some input, there exists a corresponding sequence (x_n) in I with y_n = f(x_n). Since I is compact, (x_n) has a convergent subsequence (x_k) → x for some x ∈ I. Continuity of f then transfers convergence to the outputs: f(x_k) → f(x). Therefore the subsequence (y_k) converges to y = f(x), and crucially y lies in f[I]. This matches the compactness criterion: every sequence in f[I] has a convergent subsequence whose limit stays inside f[I].

The takeaway is twofold: first, continuous images of compact sets remain compact; second, any continuous function defined on a compact set must attain its maximum and minimum. That extremum-attainment principle becomes a workhorse in later analysis arguments.

Cornell Notes

Compactness is preserved under continuous mappings: if I ⊂ ℝ is compact and f is continuous on I, then the image f[I] is compact. On the real line, compactness is equivalent to being closed and bounded, so f[I] is both bounded and closed. That forces the supremum and infimum of f[I] to lie in f[I] itself, turning them into an actual maximum and minimum. A sequential proof makes this concrete: any sequence in f[I] comes from a sequence in I, which has a convergent subsequence because I is compact; continuity then carries the limit through f, ensuring the limit remains in f[I]. The practical payoff is that continuous functions on compact sets always achieve their extrema.

Why does compactness of I imply that every sequence of inputs has a convergent subsequence with a limit still in I?

Compactness (using the sequential definition) means: for any sequence (x_n) with x_n ∈ I, there exists a subsequence (x_k) that converges to some x ∈ I. The limit cannot “leave” the set, which is exactly what makes compactness stronger than mere boundedness.

How does the proof move from a sequence in the image f[I] back to a sequence in I?

Given a sequence (y_n) in f[I], each y_n must equal f(x_n) for at least one x_n ∈ I (because y_n is in the image). This produces a corresponding sequence (x_n) in I such that y_n = f(x_n) for all n.

Where does continuity enter, and what does it guarantee about limits?

Continuity guarantees that if x_k → x, then f(x_k) → f(x). In the proof, once a subsequence (x_k) converges to x ∈ I, continuity ensures the associated subsequence (y_k = f(x_k)) converges to y = f(x), keeping the limit inside f[I].

Why does “bounded and closed” matter for turning suprema/infima into maxima/minima?

Boundedness ensures the supremum and infimum exist as real numbers. Closedness ensures those limiting values actually belong to the set. Since f[I] is compact, it is both bounded and closed, so the supremum becomes a maximum and the infimum becomes a minimum within f[I].

What can go wrong without continuity?

Without continuity, the limit of f(x_k) need not equal f(lim x_k). As a result, a sequence of function values can converge to a value that lies outside the set of attained outputs, so the supremum/infimum may fail to be achieved by any point in I.

Review Questions

State the theorem about images of compact sets under continuous functions and specify the assumptions needed.
Using the sequential definition of compactness, outline the key steps that show f[I] is compact.
Explain why a continuous function on a compact set must attain both a maximum and a minimum.

Key Points

1
If I ⊂ ℝ is compact and f: I → ℝ is continuous, then the image f[I] is compact.
2
On ℝ, compactness is equivalent to being closed and bounded (Heine–Borel).
3
Because f[I] is closed and bounded, its supremum and infimum belong to f[I], becoming a maximum and minimum.
4
A continuous function on a compact set always attains its maximum and minimum values at points in the domain.
5
The sequential proof starts with an arbitrary sequence in f[I], lifts it to a sequence in I, extracts a convergent subsequence using compactness of I, and then applies continuity to pass the limit through f.
6
Continuity is essential: without it, maxima/minima may fail to be achieved even if suprema/infima exist.

Highlights

Compactness survives continuous maps: f[I] is compact whenever I is compact and f is continuous.

Closedness plus boundedness forces extrema to be attained, not just approached.

The sequential argument is the engine: sequences in f[I] lift to sequences in I, then continuity carries limits back to the image.

A continuous function on a compact set must hit both its highest and lowest output values.

Topics

Compact Sets
Continuous Functions
Heine–Borel
Sequential Compactness
Extrema Theorem