Understanding Continuity in Mathematical Functions: Exploring the Definition and Example

the function is continuous at x=a if

The function f(x) is said to be continuous at x = a if three conditions are met:

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The function f(x) is said to be continuous at x = a if three conditions are met:

1. The function is defined at x = a, which means that f(a) is defined.
2. The limit of the function as x approaches a exists. In other words, lim(x→a) f(x) exists.
3. The limit and the value of the function are equal at x = a. This means that lim(x→a) f(x) = f(a).

If all three conditions are satisfied, then the function is continuous at x = a.

To illustrate this concept, let’s consider an example:

Let f(x) = 2x + 3. We want to determine if this function is continuous at x = 2.

1. The function is defined at x = 2 because we can substitute x = 2 into f(x) to get f(2) = 2(2) + 3 = 7.

2. To check the limit as x approaches 2, we can set up the following limit expression:
lim(x→2) (2x + 3)

Since this is a linear function, the limit can be evaluated by directly substituting x = 2:
lim(x→2) (2x + 3) = 2(2) + 3 = 7.

3. The value of the function f(x) at x = 2 is also 7. So, f(2) = 7.

Since all three conditions are met (f(2) is defined, the limit exists and equals the value of the function at x = 2), we can conclude that the function f(x) = 2x + 3 is continuous at x = 2.

This example demonstrates the criteria for continuity at a specific point. A function can be continuous at multiple points or over an interval if these conditions hold for each point within that interval.

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