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A derivative has neither a removable discontinuity nor a jump discontinuity
Statement
A derivative has neither a removable discontinuity nor a jump discontinuity. Any discontinuity of a derivative is therefore essential in the classification of Discontinuity of at a point of its domain, and its classification: removable discontinuity, jump discontinuity and essential discontinuity, equivalently Rudin's discontinuities of the first and of the second kind.
Facts & Assumptions
Given: on an interval and a discontinuity point .
The derivative has the intermediate value property (Darboux's theorem: every derivative has the intermediate-value property).
At a removable discontinuity both finite one-sided limits agree, while at a jump they are finite and unequal (Discontinuity of at a point of its domain, and its classification: removable discontinuity, jump discontinuity and essential discontinuity, equivalently Rudin's discontinuities of the first and of the second kind, The left and right limits of at , as limits of the restrictions of to and ).
Proof
Assume is removable. Choose a value strictly between the common punctured limit and . On a sufficiently small punctured neighbourhood all values of lie on the limit side of that value, while the endpoint value lies on the other side, contradicting the intermediate value property on a segment ending at .
Assume is a jump. The open interval between the unequal one-sided limits contains a value different from ; choose such a value. Sufficiently close points on the two sides have values on opposite sides of the chosen value, while neither punctured side nor the point takes it, again contradicting the intermediate value property.
Thus neither kind of discontinuity can occur.
Depends on
- Darboux's theorem: every derivative has the intermediate-value property
- Discontinuity of $f$ at a point of its domain, and its classification: removable discontinuity, jump discontinuity and essential discontinuity, equivalently Rudin's discontinuities of the first and of the second kind
- The left and right limits of $f$ at $c$, as limits of the restrictions of $f$ to $A \cap (-\infty, c)$ and $A \cap (c, \infty)$
Used by
Nothing in the library uses this result yet.
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Sources
- J. Lebl, Basic Analysis I, Taylor's theorem and related calculus (standard reference, not scraped)
- MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 lecture notes (standard reference, not scraped)
- Colgate University MATH 323, Chapter 5 notes (standard reference, not scraped)
- University of Pennsylvania, derivatives and discontinuities (standard reference, not scraped)