definition of continuity: f is continuous at c iff
If f is continuous on [a,b] and k is a number between f(a) and f(b), then there is at least one number c between a and b such that f(c)=k
Definition of continuity:
A function f is said to be continuous at point c if three conditions are met:
1. f(c) is defined, i.e., f(c) exists.
2. The limit of f(x) as x approaches c exists. Mathematically, this means that:
limit as x approaches c of f(x) = f(c)
3. The limit must be equal to the function value at c. This means that the left and right limits of the function should be equal, i.e.:
limit as x approaches c from the left of f(x) = limit as x approaches c from the right of f(x) = f(c)
If all three conditions are met, then the function is said to be continuous at c. Intuitively, this means that as x approaches c from either side, the value of f(x) approaches f(c), and there are no jumps or gaps in the function at c. If any one of the three conditions is not met, then we say that the function is not continuous at c.
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