# Epsilon-Delta I write this because people really don't understand epsilon-delta. Although I'm not good at analysis, at least I understand epsilon-d elta so... ## Continuity Let's start with continuity. Continuous means "no jump". Then what is a "jump"? ------------------------------------------------------------------ Suppose, for any d>0, there exists an e>0 s.t. |f(x+-d) - f(x)| >= e. Then we will say f has a jump at x. ------------------------------------------------------------------ By slightly changing, ------------------------------------------------------------------ Suppose, for any d>0, there exists an e>0 s.t. |f(x) - f(x0)| >= e for every x satisfying |x-x0| < d. Then we will say f has a jump a t x0. ------------------------------------------------------------------ Continuous at x means no jump at x. By negating, ------------------------------------------------------------------ Suppose, for any e>0, there exists an d>0 s.t. for all x, |x-x0| < d implies |f(x) - f(x0)| < e. Then we will say f is continuous at x0. ------------------------------------------------------------------ This is the definition of continuity. ## Limit If we use p,L instead of x0,f(x0), we get the definition of limit. This means the limit is what we get by approaching without a jump. ------------------------------------------------------------------ Suppose, for any e>0, there exists an d>0 s.t. for all x, |x-p| < d implies |f(x) - L| < e. Then we will say the limit of f, as x ap proaches p, is L. We write this as lim_{x->p} f(x) = L ------------------------------------------------------------------ For x->inf, change (p-d,p+d) with (N0,inf). We can think this as n o jump at infinity. ------------------------------------------------------------------ Suppose, for any e>0, there exists an N0 <- N s.t. for all x, x > N0 implies |f(x) - L| < e. Then we will say the limit of f, as x a pproaches inf, is L. We write this as lim_{x->inf} f(x) = L ------------------------------------------------------------------ It works for a sequence too. Because a_n = f(n). ------------------------------------------------------------------ Suppose, for any e>0, there exists an N0 <- N s.t. for all n, n > N0 implies |a_n - L| < e. Then we will say the limit of a_n is L. We write this as lim_{n->inf} a_n = L ------------------------------------------------------------------