Limits and Continuity, The Limit of a Function

The Limit of a Function

Let the function f map one metric space into another. For convenience, assume it maps reals to reals, f(x) = y. The limit of f(x) as x approaches p is l, if for every ε > 0 thereis δ > 0 such that |x,p| < δ implies |f(x),l| < ε. As x moves towards p, f(x) moves towards l.

If f(p) = l then f is continuous at p. In other words, the value of f at p equals the limit of f at p.

When x is real, one-sided limits (restricting x>p or x<p) are sometimes useful. Infinite limits, as x approaches + or - infinity, are also possible. In higher dimensions we might talk about x approaching infinity in any direction. For instance, 1/(x²+y²) approaches 0 as x and y approach infinity. To be rigorous, every ε has an r such that points in the plane, more than r units from the origin, have f(x,y) < ε. Of course the reals offer the advantage of restricting x to +∞ or -∞.