Continuity at a Point and on an Open Interval
Ever wonder if you could draw a function without lifting your pencil? That's essentially what continuity is all about! A continuous function has no holes, jumps, or gaps in its graph.
For a function to be continuous at a point c, three conditions must be met: f must be defined, the limit of f as x approaches c must exist, and that limit must equal f. If a function meets these conditions at every point within an interval (a,b), we say it's continuous on that interval.
Not all functions are continuous everywhere. For example, f = 1/x isn't continuous at x = 0 because division by zero isn't defined. Piecewise functions might have discontinuities where the pieces meet, and rational functions can have issues where the denominator equals zero.
Remember This: Discontinuities come in two flavors - removable (where we could "fix" the function by redefining it at a single point) and nonremovable (where no single value can make the function continuous at that point).
When checking a function for continuity, always work through all three conditions systematically. Many functions you'll encounter in basic calculus (like polynomials and sine/cosine) are continuous everywhere, which is why they're so useful for modeling real-world situations.
