Real functions Limits and continuity Example

Thomae's function

Builds on Continuity at a point Sequential characterization of limits

Example

Define $f \colon \mathbb{R} \to \mathbb{R}$ as follows. For irrational $x$ , $f(x) = 0$ . For rational $x = p/q$ in lowest terms with $q > 0$ , $f(x) = 1/q$ . Then $f$ is continuous at each irrational number and discontinuous at each rational number.

Remarks

Let $a$ be irrational and let $\varepsilon > 0$ . Only finitely many $p/q$ near $a$ have $1/q \ge \varepsilon$ . So $f(x) < \varepsilon$ for all $x$ close enough to $a$ . At a rational $a$ , irrational points near $a$ have value $0$ , but $f(a) > 0$ . The function is periodic with period $1$ . No function on $\mathbb{R}$ is continuous exactly at the rationals.

Reading path

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Limit point of a set definition
Sequence definition
Limit of a function definition
Real sequence definition
Continuity at a point definition
Limit of a sequence definition
Sequential characterization of limits statement
Thomae's function example

Tag	`0HJ28H9Y95`
URI	`m2://0HJ28H9Y95`
Aliases	`popcorn-function`
Kind	example
Writing	W0 stub
MSC	26A15
Uses	Continuity at a point, Sequential characterization of limits
Same as	Wikidata Q776401 Wikipedia Thomae's function

Thomae's function

Example

Remarks

Reading path

Further reading

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