Differentiation is one of the most important fundamental operations in calculus. Its theory primarily depends on the idea of limit and continuity of function.
We know that for every point, there exists neighboring points.
But when we say that the function f(x) is continuous at a point x=a, we mean that at point ( a, f(a) ) the graph of the function f(x) has no holes or gaps or discontinuity. That is, its graph is unbroken at point (a, f(a))