# ($\varepsilon$, $\delta$)-definition of limit by Weierstrass

I am looking for the original ($$\varepsilon$$, $$\delta$$)-definition of limit by Weierstrass, but I cannot find an exact quote or a reference. I saw that somewhere it was claimed that this definition was actually published in Heine's notes of Weierstrass' lectures, but I did not find any exact reference either.

Where could I check the exact formulation of Weierstrass' definition? (I guess I know the definition, I just want to be sure that this is how he formulated it.)

(To explain my interest, I am trying to find out who was that evil person who decided to use the condition $$0 < |x - a| <\delta\Rightarrow|f(x) - b| <\varepsilon$$ instead of $$|x - a| <\delta\Rightarrow|f(x) - b| <\varepsilon$$.)

• I cannot be sure, but it is quite possible that Weierstrass never published it. The problem with Weierstrass was that many of his results were included in his famous lectures, and he never published them in journals. Lectures were quite popular, some have been recorded by students and spread, but they were never formally published. – Alexandre Eremenko Mar 14 at 23:18
• Isn't it true that $(\epsilon,\delta)$ definition of a limit was invented by Cauchy? – user14095 Mar 15 at 22:49
• @dodd, i am not an expert, but i've read that Cauchy did not explain the relation between $\varepsilon$ and $\delta$. – Alexey Mar 15 at 23:11
• I was taught that he invented that in the first analysis course 48 years ago. I never thought anybody would doubt that fact. – user14095 Mar 15 at 23:15
• @DaveLRenfro: It is correct. 48 years ago I was a student in an Analysis class. What is your problem? – user14095 Mar 16 at 15:16

The issue is that most of W's foundational works was unpublished or scattered into his lectures.

See Pierre Dugac, Eléments d'analyse de Karl Weierstrass (AHES, 1973), Appendice II, page 119, with some extracts from Ausarbeitung der Vorlesung an dem Konig. Gewerbeinstitut zu Berlin im Sommersem. 1861 von H.A. SCHWARZ (Institut Mittag-Leffler)

Ist $$f(x)$$ eine Funktion von $$x$$ und $$x$$ ein bestimmter Wert, so wird sich die Funktion, wenn $$x$$ in $$x + h$$ ubergeht, in $$f(x + h)$$ andern; die Differenz $$f(x + h) f(x)$$ nennt man die Veranderung, welche die Funktion dadurch erfahrt, dass das Argument von $$x$$ in $$x + h$$ ubergeht. Ist es nun moglich, fur heine Grenze $$\delta$$ zu bestimmen, sodass fur alle Werte von $$h$$, welche ihrern absoluteu Betrage noch kleiner als $$\delta$$ sind, $$f(x + h) - f(x)$$ kleiner werde als irgendeine noch so kleine Grosse $$\epsilon$$,

• This is a definition of continuity, but i am looking for his definition of limit. – Alexey Mar 15 at 11:52