Continuity and Differentiability | Class 12 Mathematics Notes

Continuity of a Function

Continuity of a function at a point is a fundamental concept in calculus that describes how a function behaves as the input approaches that point. Formally, a function f(x) is said to be continuous at a point x = a if three conditions are satisfied: (1) f(a) is defined, meaning the function has a value at x = a; (2) the limit of f(x) as x approaches a exists; and (3) the limit of f(x) as x approaches a is equal to f(a). This means there is no sudden jump, break, or hole in the graph of the function at x = a. If any of these conditions fail, the function is said to be discontinuous at that point.

The concept of continuity can be extended to an interval. A function is continuous on an interval if it is continuous at every point in that interval. Continuity is crucial in calculus because many theorems, such as the Intermediate Value Theorem, require functions to be continuous.

There are different types of discontinuities: removable discontinuity (where the limit exists but is not equal to the function value), jump discontinuity (where left-hand and right-hand limits exist but are not equal), and infinite discontinuity (where the function approaches infinity).

The NCERT textbook introduces the concept with precise definitions and examples to illustrate continuous and discontinuous functions. It also discusses how to check continuity using limits and function values.

📊 Diagram: The NCERT textbook includes diagrams showing graphs of continuous functions like y = x², and graphs with removable and jump discontinuities illustrating the concepts visually.

🧪 Activity: Activity: Students are asked to verify continuity of given functions at specified points by evaluating limits and function values.

🔗 Connection: This section lays the foundation for understanding continuity in composite functions, which is explored in the next section.