Can we rephrase Epsilon-delta definition?

Question

Can we rephrase Epsilon-delta definition like this: "If for any positive number δ (delta) there exists a positive number ε (epsilon) such that if f(x) is within ε units of L (i.e., |f(x) - L| < ε), then x is within δ units of c (i.e., if |x - c| < δ)." If not then why can't this definition be used, can anyone please comment on my doubt.

Answer 1

No:

1. Your definition fails to account for the requirement that every $x$ in a (punctured) neighborhood of $c$ must satisfy $|f(x) - L| < \epsilon$. A counterexample that has a limit but does not satisfy your definition is the constant function $f(x) = 0$. The limit as $x \to 0$ is $L = 0$, and we have $|f(x) - L| = 0 < \epsilon$ regardless of the value of $x$ or $\delta$.
2. You relate $\epsilon$ to $\delta$ but do not require that $\epsilon$ can be as small as one pleases, so a function can fail to have a limit but still meet your definition. One such counterexample is the step function $$f(x) = \begin{cases}1, & x \ge 0 \\ 0, & x < 0. \end{cases}$$ This function is bounded, so $$|f(x) - L| \le |L| + 1$$ by the triangle inequality; thus your definition permits you to choose $\epsilon > |L| + 1$ for whichever $L$ you please. Yet the limit as $x \to 0$ obviously does not exist.