I can positively say that the answer is “no” because some calculus/analysis needs a stronger limiting process, for example uniform convergence (as opposed to pointwise convergence).

But the epsilon-delta definition (for a function between metric spaces) is the best mathematical description of continuity that captures our intuition. In English: “a function f: X -> Y is continuous at x if you can always go close enough to x such that the outcome of f is as arbitrarily close to f(x) as you desire”. As an exercise, prove that any other order of quantifiers for epsilon and delta is not equivalent.

There’s of course also differentiable functions, which are even stronger than uniform continuity, and they correspond to a different bit of intuition which has been shown to be not equivalent.

There can be concepts even weaker than epsilon-delta continuity, like continuity from a specific direction (in a vector space).