Properties of Continuous Functions A function f(x) is continuous at a point x = a if the following three conditions are satisfied: 1. One-sided limits:...
Properties of Continuous Functions
A function f(x) is continuous at a point x = a if the following three conditions are satisfied:
One-sided limits: lim_{x->a^-} f(x) = lim_{x->a^+} f(x).
Two-sided limits: lim_{x->a} f(x) = f(a).
Limit at a: f(a) = f(a).
If these conditions are satisfied, then the limit of f(x) as x approaches a is equal to f(a).
Examples:
A function f(x) = |x| is continuous at x = 0 because both left and right limits are equal to f(0) = 0.
A function f(x) = x^2 is continuous at x = 0 because the one-sided limits and two-sided limits are equal to f(0) = 0.
A function f(x) = 1/x is not continuous at x = 0 because the limit on the left side is undefined while the limit on the right side is equal to 1.
Implications of Continuity:
Continuity has several important properties that make functions well-behaved. These properties allow us to make predictions about the function's behavior near a given point, and we can use them to evaluate the function at a given point.
Continuity guarantees that the function has a definite derivative at a point.
Continuity guarantees that the function has a definite definite integral from a to b.
Continuity guarantees that the function is differentiable at a point.
Continuity guarantees that the function is continuous on an interval.