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Continuity of Functions

Continuity of Functions

Continuity at a Point

Continuity is easiest if we begin by thinking of it at a single point. Once we have that down we can start thinking of continuity in broader terms. There's a couple conditions that have to be met for us to say a function is continuous at a point c.

The first condition is that f(c) has to actually exist. We can't have a hole in the graph at c, or an asymptote, or anything that's going to make f(c) not exist as a nice, real number.

In other words, c has to be in the domain of f.

This isn't the only condition, though. We also need

.

If these two conditions are met, we say that f is continuous at x = c.

In words, the function doesn't jump around at x = c. There will be no surprises; the function will pass smoothly through x = c unscathed. The limit as we approach c will exist and be equal f(c).

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