definition of continuity: f is continuous at c iff
the limit of f(x) as x approaches c exists and is equal to f(c)
the limit of f(x) as x approaches c exists and is equal to f(c).
The formal definition of continuity states that a function f is continuous at a point c if three conditions are met:
1. The function must be defined at c: This means that f(c) must be defined, or in other words, there should be a value for the function at c.
2. The limit of f(x) as x approaches c must exist: This means that as x gets closer and closer to c, the values of f(x) approach a certain value. In mathematical terms, the limit of f(x) as x approaches c must be a real number.
3. The limit value must be equal to f(c): This means that the value of the limit as x approaches c must be the same as the value of f(c). In other words, the function must not have any sudden jumps or breaks at c.
If all these conditions are satisfied, then the function f is said to be continuous at the point c. Continuity can also be defined for intervals or the entire domain of a function.
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