Understanding Continuity in Mathematical Functions | Explained and Illustrated

f is cont at x=a

If a function f is continuous at x=a, it means that the function is defined at a, and the limit of the function as x approaches a exists and is equal to the value of the function at a

If a function f is continuous at x=a, it means that the function is defined at a, and the limit of the function as x approaches a exists and is equal to the value of the function at a.

In mathematical terms, we can express the continuity of f at x=a using the following conditions:

1. f(a) is defined: This means that the function is evaluated at x=a and has a well-defined value.

2. The limit of f(x) as x approaches a exists: The function should have a limit as x approaches a. This implies that as x gets arbitrarily close to a (but not necessarily equal to a), the values of f(x) do not drastically change.

3. The value of the limit equals f(a): The limit of f(x) as x approaches a should equal the value of f(a). This implies that the function is “continuous” at the point x=a since the function value and the limit value coincide.

Overall, if these three conditions are satisfied, f is continuous at x=a. This means that there are no holes, jumps, or gaps in the graph of the function at x=a. The function can be smoothly traced without any disruptions.

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