Understanding Continuity in Mathematics: Explained with Conditions and Definitions

What does it mean for a function f(x) to be continuous at a point x = a in its domain?

When a function f(x) is said to be continuous at a point x = a in its domain, it means that the function is well-behaved and does not have any abrupt changes or jumps at that specific point

When a function f(x) is said to be continuous at a point x = a in its domain, it means that the function is well-behaved and does not have any abrupt changes or jumps at that specific point. In other words, there are no holes, gaps, or vertical asymptotes in the graph of the function at x = a.

To formally define continuity at a point, we need to consider three main conditions:

1. The function f(a) is defined at x = a. The value of the function at x = a must exist, meaning that it is not undefined or involves division by zero.

2. The limit of f(x) as x approaches a exists. The left-hand limit and the right-hand limit of the function as x approaches a must both be finite and equal. This means that as we approach x = a from both sides (left and right), the function approaches the same value.

lim f(x) = lim f(x)
x→a- x→a+

3. The value of the function at x = a is equal to the limit. The value of the function at x = a must be equal to the limit value.

f(a) = lim f(x)
x→a

If all three conditions are met, then the function f(x) is continuous at x = a. This means that the graph of the function is a continuous, smooth curve without any interruptions or breaks at the point x = a.

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