The => direction is pretty simple if you understand the definitions. For the converse, try a proof by contradiction. Assume that f is not continuous at c and consider ε for which there is no δ from the definition. In particular, δ = 1, 1/2, 1/3, ..., 1/n, ... do not work for this ε. For each natural n, choose x_n that is a witness that δ = 1/n "does not work" for this ε. Then the sequence x_n will violate the assumed property (that if \lim_{n \to \infty} x_n = c, then \lim_{n \to \infty} f(x_n) = f(c)).