Lipschitz & One-Insight Problems

Every year, GATE slips in a calculus question that looks brutal on first read. An abstract function f, no formula, a strange-looking inequality, and a request for something specific. There’s nothing to differentiate, nothing to plug in. It seems unfair.

It isn’t. These problems are one-trick ponies. Each one hinges on a single insight that, once you see it, collapses the whole thing to a one-line answer. This lesson teaches the most famous example — and, more usefully, how to spot the species.

The star concept here, Lipschitz bounds (a cap on how fast a function can change), is not just exam trivia: bounding a function’s or gradient’s Lipschitz constant is how modern ML keeps training stable — it underpins WGAN’s weight clipping, certified robustness bounds, and convergence guarantees for gradient descent.

The signature problem: a squared bound

Here is the setup that has appeared more than once. You are told a function satisfies

|f(x) − f(y)| ≤ C · (x − y)²      for all real x, y

for some constant C. There is no formula for f. The question asks for something like f(1) − f(0). It looks underspecified — but the squared (x − y)² is doing all the work.

The one idea: divide, then take the limit

The derivative of f at a point x is the limit of the difference quotient. So look at that quotient and feed it the bound. For y ≠ x, divide both sides by |x − y|:

| f(x) − f(y) |
| ----------- |  ≤  C · |x − y|
|    x − y    |

The left side is the difference quotient whose limit (as y → x) is |f'(x)|. The right side C·|x − y| goes to 0 as y → x. Squeezed between, the slope has nowhere to go:

0 ≤ |f'(x)| ≤ lim (C·|x − y|) = 0      ⇒      f'(x) = 0   for every x

A function whose derivative is zero everywhere is constant. So f is the same value at every point, and therefore any difference vanishes:

f'(x) = 0 everywhere   ⇒   f is constant   ⇒   f(1) − f(0) = 0

How GATE asks this

Almost always a NAT whose answer is a clean 0. The wrapper changes — it might ask for f(2) − f(−3), or “how many such non-constant f exist?”, or give the bound as |f(x) − f(y)| ≤ 5(x − y)² — but the engine is identical: divide by |x − y|, let y → x, conclude f' ≡ 0, so f is constant and the difference is 0. The skill GATE is testing is recognition: an abstract f plus a squared (or higher power) bound is the tell.

Worked example — a real GATE DA question

A function f satisfies |f(x) − f(y)| ≤ (x − y)² for all real x, y. Find f(1) − f(0).

Run the pattern. Fix x, take any y ≠ x, and divide the given bound by |x − y|:

| f(x) − f(y) |
| ----------- |  ≤  |x − y|
|    x − y    |

Now let y → x. The left side tends to |f'(x)| (the definition of the derivative), and the right side |x − y| → 0. By the squeeze,

|f'(x)| ≤ 0   ⇒   f'(x) = 0      for every x.

So f has zero slope everywhere and is therefore constant. A constant function takes the same value at 1 and at 0, hence

f(1) − f(0) = 0.

The answer is 0. This is a real GATE DA NAT (2024/2025 style) — daunting on first read, a single line once you spot that the squared bound kills the derivative.

Quick check