Understanding Continuity in Mathematics: Conditions and Definitions

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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