definition of continuity: f is continuous at c iff
the following three conditions are satisfied:
1
the following three conditions are satisfied:
1. f(c) is defined: The value of the function f at the point c is well-defined, meaning there are no gaps or undefined values at that point.
2. The limit of f(x) as x approaches c exists: The behavior of the function as x gets arbitrarily close to c is predictable and does not have sudden jumps or discontinuities.
3. The limit of f(x) as x approaches c is equal to f(c): The value of the function at the point c is the same as the limit of the function as x approaches c. This indicates that there are no abrupt changes or abrupt jumps in the function’s behavior at the point c.
In simpler terms, a function is continuous at a point c if the function is defined at that point, the function approaches a predictable value as x gets very close to c, and this predictable value is equal to the value of the function at point c. If any of these conditions is not met, then the function is said to be discontinuous at that point.
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