The
definition of continuity at a point is a formal way to express the intuitive idea that a function can be drawn without lifting your pen from the paper. Formally, a function
f is continuous at a point
c if the following three conditions are met:
- The function f(x) is defined at c, which means f(c) exists.
- The limit of f(x) as x approaches c exists.
- The limit of f(x) as x approaches c is equal to f(c).
The expression
\[\lim_{x \to c} f(x) = f(c)\]
is a succinct way to capture all three of these conditions into a single equation, symbolizing the smooth passing of the function through the point
c.