Properties of Continuous Functions on Compact Sets

At a Glance

• Frequency: 1 sub-part across 1 of 13 years (2017)
• Priority tier: T4
• Marks (count): 10 (1)
• Average solve time: ~12 min
• Difficulty mix: medium 1
• Section: A | Dominant type: computation/proof

Why This Chapter Matters

This is a T4 atom with a single appearance (2017, 10 marks). The questions are typically short — either prove the Extreme Value Theorem, apply the Intermediate Value Theorem to locate a root, or show that a continuous function on $[a,b]$ is bounded. The results themselves are classical theorems most candidates know by name; UPSC tests whether you can write a clean, self-contained argument.

Minimum Theory

Compact subsets of $\mathbb{R}$ (Heine-Borel). A subset $K \subseteq \mathbb{R}$ is compact if and only if it is closed and bounded.

Theorem 1 (Extreme Value Theorem). If $f:[a,b]\to\mathbb{R}$ is continuous, then $f$ attains its maximum and minimum on $[a,b]$; i.e., there exist $c, d \in [a,b]$ such that

$f(c) \leq f(x) \leq f(d) \quad \text{for all } x \in [a,b].$

Proof idea. $f$ is bounded because $[a,b]$ is compact and continuous images of compact sets are compact (hence bounded). Let $M = \sup_{[a,b]} f$. There exists a sequence $x_n \in [a,b]$ with $f(x_n) \to M$. By Bolzano-Weierstrass, $(x_n)$ has a subsequence $x_{n_k} \to c \in [a,b]$. By continuity, $f(c) = M$.

Theorem 2 (Intermediate Value Theorem). If $f:[a,b]\to\mathbb{R}$ is continuous and $f(a) < \gamma < f(b)$ (or $f(b) < \gamma < f(a)$), then there exists $c \in (a,b)$ with $f(c) = \gamma$.

Corollary. $f([a,b])$ is a closed bounded interval $[m, M]$.

Theorem 3 (Uniform continuity on compact sets). If $f:[a,b]\to\mathbb{R}$ is continuous, then $f$ is uniformly continuous on $[a,b]$; i.e., for every $\varepsilon > 0$ there exists $\delta > 0$ (depending only on $\varepsilon$, not on $x$) such that

$|x - y| < \delta \;\Rightarrow\; |f(x) - f(y)| < \varepsilon \quad \text{for all } x, y \in [a,b].$

Theorem 4 (Compactness is preserved). If $K \subseteq \mathbb{R}$ is compact and $f:K\to\mathbb{R}$ is continuous, then $f(K)$ is compact.

IVT application — locating roots. To show $g(x) = 0$ has a solution in $(a,b)$: compute $g(a)$ and $g(b)$; if they have opposite signs, IVT guarantees a root.

Question Archetypes

ivt-root (1 question(s); 2017)

Recognition Cues

• An equation $f(x) = 0$ or $f(x) = x$ (fixed point) on a closed interval.
• The function is continuous; signs of $f$ at the endpoints are opposite or can be checked.

Solution Template

1. Verify $f$ is continuous on $[a,b]$.
2. Compute or estimate $f(a)$ and $f(b)$.
3. Show $f(a)$ and $f(b)$ have opposite signs (or one is zero).
4. Apply IVT: there exists $c \in (a,b)$ with $f(c) = 0$.
5. If uniqueness is needed, argue monotonicity.

Worked Example

2017 Paper 2, 2017-P2-Q3c (10 marks)

Let $f:[0,1]\to[0,1]$ be a continuous function. Prove that $f$ has a fixed point, i.e., there exists $c \in [0,1]$ such that $f(c) = c$.

Proof.

Define $g:[0,1]\to\mathbb{R}$ by $g(x) = f(x) - x$.

Continuity. Since $f$ is continuous on $[0,1]$ and $x \mapsto x$ is continuous, $g$ is continuous on $[0,1]$.

Sign check at endpoints. Since $f:[0,1]\to[0,1]$, we have $f(0) \geq 0$, so

$g(0) = f(0) - 0 = f(0) \geq 0.$

Similarly $f(1) \leq 1$, so

$g(1) = f(1) - 1 \leq 0.$

Case 1. If $g(0) = 0$, take $c = 0$; if $g(1) = 0$, take $c = 1$. Done.

Case 2. If $g(0) > 0$ and $g(1) < 0$, then by the Intermediate Value Theorem (which applies since $g$ is continuous on the closed interval $[0,1]$ and $g(1) < 0 < g(0)$), there exists $c \in (0,1)$ such that $g(c) = 0$, i.e., $f(c) = c$.

In every case, $f$ has a fixed point in $[0,1]$.

$\boxed{f(c) = c \text{ for some } c \in [0,1].}\;\blacksquare$

Common Traps

• Forgetting to handle the boundary cases $g(0) = 0$ or $g(1) = 0$ separately before invoking IVT (IVT requires a strict inequality between the endpoint values).
• Not citing that $f:[0,1]\to[0,1]$ forces $f(0) \geq 0$ and $f(1) \leq 1$; you must derive the sign conditions explicitly.
• Confusing IVT (existence of a $c$ with $f(c) = \gamma$) with the Extreme Value Theorem (attainment of max/min); for a fixed-point question, IVT is the right tool.

Marks-Aware Writing

At 10 marks (Section A), be precise but not verbose. The expected structure is: define $g$, establish continuity (1 sentence), compute $g(0)$ and $g(1)$ with reasoning, handle boundary cases, apply IVT, conclude. This should fill about half a page. Avoid lengthy preliminaries about compactness theory — the question tests IVT application directly.

Practice Set

Only one historical question on this atom (shown above).