I have written the question below verbatim:
Suppose a function $f : \mathbb { R } \rightarrow \mathbb { R }$ satisfies the following property: $$\forall \varepsilon > 0 , \exists \delta > 0 , \forall x , y \in \mathbb { R } , \quad | x - y | < \delta \Rightarrow | f ( x ) - f ( y ) | < \varepsilon$$
(in words: for any positive number $\varepsilon$, there exists a positive number $\delta$ such that for any pair of real numbers $x, y$ the inequality $| x - y | < \delta$ implies $ f ( x ) - f ( y ) | < \varepsilon$)
Does it then follow that $f$ is continuous on $\mathbb { R }$?
However, this simply looks to be the $\varepsilon - \delta$ definition of continuity to me. Is there any subtle difference that I am missing here? If so, what would the answer be?