Understanding Continuity in Mathematics: Definition and Conditions

definition of continuity: f is continuous at c iff

Definition of continuity: A function f is said to be continuous at a point c in its domain if the following three conditions hold:

1

Definition of continuity: A function f is said to be continuous at a point c in its domain if the following three conditions hold:

1. The function f(c) is defined (i.e., the value of the function at c exists).
2. The limit of f(x) as x approaches c exists. This means that the left-hand limit and the right-hand limit of f(x) as x approaches c are equal. Mathematically, it can be expressed as:

lim(x->c-) f(x) = lim(x->c+) f(x) = f(c)

The left-hand limit, lim(x->c-) f(x), represents the behavior of the function as x approaches c from the left side, while the right-hand limit, lim(x->c+) f(x), represents the behavior of the function as x approaches c from the right side.

3. The value of the limit in condition 2 is equal to the value of the function at c. Mathematically, it can be expressed as:

lim(x->c) f(x) = f(c)

This condition ensures that there are no “jumps” or “holes” in the graph of the function at the point c.

If all three conditions are satisfied, then the function is continuous at the point c. It means that the function can be drawn without lifting the pen from the paper at the point c, and there is a smooth, unbroken curve passing through the point (c, f(c)).

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