Continuity and Differentiability | Class 12 Mathematics Notes

Continuity of Composite Functions

Composite functions are formed by applying one function to the result of another function, denoted as (f ∘ g)(x) = f(g(x)). The continuity of composite functions depends on the continuity of the individual functions involved.

The NCERT textbook explains that if g is continuous at x = a and f is continuous at g(a), then the composite function f(g(x)) is continuous at x = a. This is because the limit of g(x) as x approaches a exists and equals g(a), and since f is continuous at g(a), the limit of f(g(x)) as x approaches a equals f(g(a)).

This property is important because it allows us to build complex continuous functions from simpler continuous functions. The textbook provides examples such as f(x) = sin(x²), where the inner function g(x) = x² is continuous everywhere, and the outer function f(u) = sin u is continuous everywhere, so their composition is continuous everywhere.

The section also discusses how to verify continuity of composite functions by checking the continuity of the inner and outer functions at the appropriate points.

📊 Diagram: The textbook shows graphs of composite functions illustrating continuity, including plots of f(g(x)) where both f and g are continuous functions.

🧪 Activity: Activity: Students verify continuity of composite functions by evaluating limits and function values for given f and g.

🔗 Connection: Understanding continuity of composite functions is essential before studying differentiability, which is the next section.