Continuity of a Function at a Point A function is continuous at a point if the limit of the function as the independent variable approaches that point is eq...
Continuity of a Function at a Point
A function is continuous at a point if the limit of the function as the independent variable approaches that point is equal to the function value at that point. In other words, the function takes on the same value as the limit when the input approaches the input.
Example:
Consider the function (f(x) = \frac{x}{x}) at the point (x = 0). The limit of this function as (x) approaches (0) is equal to 1, which is equal to the function value at (x = 0). Therefore, (f(x)) is continuous at (x = 0).
Continuity of a Function in an Interval
A function is continuous in an interval if it is continuous at every point within that interval. In other words, the function takes on the same values within that interval as it does at any other point within that interval.
Example:
Consider the function (f(x) = x^2) in the interval ([0, 4]). Since this function is continuous for all (x) in the interval, it is continuous in the interval ([0, 4])