the function is continuous at x=a if
In mathematics, a function is said to be continuous at a point “a” if three conditions are met:
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In mathematics, a function is said to be continuous at a point “a” if three conditions are met:
1. The function is defined at “a”.
This means that the value of the function at “a” is well-defined and does not have any gaps or undefined points.
2. The limit of the function as x approaches “a” exists.
This means that as we approach “a” from both the left and the right sides, the function values are approaching a common value. In other words, the left-hand limit and the right-hand limit of the function at “a” are equal.
Mathematically, this condition can be expressed as:
lim(x->a-) f(x) = lim(x->a+) f(x)
3. The value of the function at “a” is equal to the limit.
This means that the function value at “a” itself matches the limit of the function as x approaches “a”.
Mathematically, this condition can be expressed as:
f(a) = lim(x->a) f(x)
If all three conditions are satisfied, then the function is considered continuous at “a”.
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