what does this sign represent in math? and what does it do?

Question

Answer 1

The Greek lowercase epsilon ε, the lunate epsilon symbol ϵ, or the Latin lowercase epsilon ɛ (see above) is used in a variety of places: In mathematics (particularly calculus), an arbitrarily small positive quantity is commonly denoted ε; see (ε, δ)-definition of limit.

Answer 2

Since you included the second symbol, we may distinguish which "epsilon" is intended.

The lunate epsilon (ϵ) is not to be confused with the set membership symbol (∈).  The symbol , first used in set theory and logic by Giuseppe Peano and now used in mathematics in general for set membership ("belongs to") did, however, evolve from the letter epsilon, since the symbol was originally used as an abbreviation for the Latin word "est". In addition, mathematicians often read the symbol as "element of", as in "1 is an element of the natural numbers" for , for example. As late as 1960, itself was used for set membership, while its negation "does not belong to" (now ) was denoted by (epsilon prime).