Continuity of a function at a point is a fundamental concept in real analysis that determines whether a function can be differentiated at that point. A function...
Continuity of a function at a point is a fundamental concept in real analysis that determines whether a function can be differentiated at that point. A function is continuous at a point if its limit at that point is equal to the function value at that point. This means that the function can be approximated by a first-degree polynomial near that point.
In other words, the function behaves smoothly and continuously at that point. A function is continuous at a point if the limit of the difference quotient as the difference approaches 0 is equal to the limit of the function value as the difference approaches 0.
For example, consider the function (f(x) = \frac{1}{x}) for (x \neq 0). Since the limit of the difference quotient as (x) approaches 0 is 1 and the function value at (x = 0) is also 1, the function is continuous at (x = 0).
Continuity in an interval involves the function's ability to take any real value in that interval. A function is continuous in an interval if it is continuous at every point in that interval.
For example, consider the function (f(x) = x^2) for (x \geq 0). This function is continuous in the interval ([0, \infty)) because it is continuous at every point in that interval