x^2 is not uniformly continuous on the real line

Builds on Uniform continuity Continuous function

Counterexample

The function $f \colon \mathbb{R} \to \mathbb{R}$ with $f(x) = x^2$ is continuous but not uniformly continuous. For $n \ge 1$ , let $x_n = n$ and $y_n = n + 1/n$ . Then $|x_n - y_n| = 1/n \to 0$ , but $f(y_n) - f(x_n) = 2 + \frac{1}{n^2} > 2.$ So no $\delta$ works for $\varepsilon = 2$ .

Remarks

On each bounded interval $[-c, c]$ , the function $x^2$ is Lipschitz with constant $2c$ . So it is uniformly continuous there. The problem is that the slope $2x$ is not bounded on $\mathbb{R}$ .

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Limit point of a set definition
Limit of a function definition
Continuity at a point definition
Continuous function definition
Epsilon-delta definition of continuity statement
Uniform continuity definition
x^2 is not uniformly continuous on the real line counterexample

Tag	`0A8JXHYTJC`
URI	`m2://0A8JXHYTJC`
Kind	counterexample
Writing	W0 stub
MSC	26A15
Uses	Uniform continuity, Continuous function

x^2 is not uniformly continuous on the real line

Counterexample

Remarks

Reading path

Further reading

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