Continuity and Unlimited Error Control Limits and continuity in calculus may be described geometrically, that is, intuitively and informally, or more precisely in terms of say epsilons and deltas. The answer might be that x must agree with A to m decimal places. In some computational problems, this answer for a specified number k of decimal places may be all that is needed. But in other situations, we want in practice or in principle, unlimited error control. Unlimited error control offers motivation and a perspective on the discussion of continuity.
Continuity here represents the concept of unlimited error control in decimal computations.
Both conditions are equivalent. Each implies the other. Both conditions are equivalent. Each implies the other. Again, why depends on how you think of the real numbers.