Understanding the Limit Definition of Continuity | Exploring the Mathematical Conditions for Function Continuity

Limit Definition of Continuity

The limit definition of continuity is a mathematical concept that explains the conditions under which a function is considered continuous at a specific point

The limit definition of continuity is a mathematical concept that explains the conditions under which a function is considered continuous at a specific point. It is defined using the concept of limits.

Formally, a function f(x) is continuous at a point c if three conditions are satisfied:

1. f(c) is defined: This means that the function f(x) must have a value assigned at the point c.

2. The limit of f(x) as x approaches c exists: This implies that the left-hand limit and the right-hand limit of the function at c exist and are equal.

Mathematically, this condition can be represented as:
lim(x → c-) f(x) = lim(x → c+) f(x)

In simpler terms, the behavior of the function as x approaches c from the left (x → c-) and the behavior of the function as x approaches c from the right (x → c+) have to be consistent.

3. The limit of the function f(x) as x approaches c is equal to f(c): This means that the value of the function as x approaches c should be equal to the value of the function at c.

Mathematically, this condition can be represented as:
lim(x → c) f(x) = f(c)

If these three conditions are met, then the function is said to be continuous at the point c. This definition can be extended to state that a function is continuous on an interval when it is continuous at every single point within that interval.

It is worth noting that the limit definition of continuity is just one way to define continuity. There are other equivalent definitions, such as the sequential definition and the ε-δ definition, which provide different perspectives on this fundamental concept in mathematics.

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