Understanding Continuity in Functions: Exploring the Absence of Jumps and Breaks

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 continuous at a point x = a in its domain, it means that the function does not have any sudden jumps, breaks, or holes at that specific point

When a function f(x) is continuous at a point x = a in its domain, it means that the function does not have any sudden jumps, breaks, or holes at that specific point. In other words, there are no interruptions or discontinuities in the graph of the function at that particular point.

To understand this concept, we need to know the definition of continuity. A function is said to be continuous at a point x = a if the following three conditions are satisfied:

1. The function f(a) is defined: This means that the function has a value assigned to it at x = a. There should not be any undefined or missing values at that point.

2. The limit of the function as x approaches a exists: This means that as we approach x = a from both the left and right sides, the function’s values approach a common limit. In mathematical notation, it can be written as:
lim (x→a⁻) f(x) = lim (x→a⁺) f(x)

3. The limit of the function as x approaches a is equal to the function value at a: This means that the limit of the function as x approaches a from both directions is equal to the value of the function at x = a. In mathematical notation, it can be written as:
lim (x→a) f(x) = f(a)

If these three conditions are met, then we can say that the function f(x) is continuous at x = a. This implies that the graph of the function is a continuous curve without any breaks or jumps at that particular point.

It’s important to note that continuity is not limited to just individual points but can also hold for intervals or the entire domain of the function. A function is said to be continuous on an interval if it is continuous at every point within that interval.

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