continuous
A function is continuous at a point if its limit as x approaches that point exists and is equal to the function's value at that point. For a function to be continuous on an interval, it must be continuous at every point within that interval, and if it is continuous at every point in its domain, it is simply referred to as continuous. Continuous functions are particularly advantageous because the limit of such a function at a point can be determined by directly evaluating the function at that point.
Interpretation
A function is continuous at a point if the function's value at that point is defined, the limit of the function as it approaches that point exists, and the value of the function at that point matches the value of the limit. This ensures that the graph of the function does not have a hole or jump at that point, allowing for a smooth and unbroken transition in the function's behavior.
Applications
In calculus, the differentiability of a function at a point implies its continuity at that point, establishing a fundamental relationship between these two properties. A function may be continuous at a point without being differentiable there, as demonstrated by cases where the graph exhibits a sharp corner or cusp.