What makes a function continuous at a point?

madeeha1d8 2022-09-23 Answered
What makes a function continuous at a point?
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Answers (1)

embraci4i
Answered 2022-09-24 Author has 10 answers
Let f ( x ) be a function defined in an interval ( a , b ) and x 0 ( a , b ) a point of the interval.
Then the definition of continuity is that the limit of f ( x ) as x approaches x 0 equals the value of f ( x ) in x 0 .
In symbols:
lim x x 0 f ( x ) = f ( x 0 )
Based on the formal definition of limit, then, for every number ε > 0 we can find δ ε > 0 such that:
| x - x 0 | < δ ε | f ( x ) - f ( x 0 ) | < ε
This means that as x gets closer and closer to x 0 also f ( x ) gets closer and closer to f ( x 0 ) and thus the function is "smooth".
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