A function is continuous at a point x=ax=ax=a if it meets three key conditions: (1) the function value f(a)f(a)f(a) exists, (2) the limit of the function as xxx approaches aaa exists, and (3) the limit of the function as xxx approaches aaa is equal to the function value at aaa, that is lim⁡x→af(x)=f(a)\lim_{x\to a}f(x)=f(a)limx→a​f(x)=f(a).

Explanation of Continuity

• A function’s graph is continuous if it can be drawn without lifting the pencil, meaning no breaks, jumps, or holes exist at that point or over an interval.

• Mathematically, continuity ensures that as inputs get arbitrarily close to a particular point, the outputs of the function get arbitrarily close to the function's value at that point.

Continuity over intervals

• A function is continuous over an interval if it is continuous at every point in that interval.

• Common examples of continuous functions over their domains include polynomials, exponential functions, sine, and cosine functions.

Thus, the essence of a function being continuous is no interruptions or abrupt changes in value at the point or throughout an interval, formalized by the equality of the function value and the limit approaching that value.