Continuity and Differentiability — Class 12 Mathematics

This chapter formalises continuity and differentiability. A function is continuous at if ; it is continuous on an interval if continuous at every point. Differentiability is stronger: exists if the limit defining the derivative exists, and differentiability implies continuity (but not conversely). The chapter develops rules for differentiating composite functions (chain rule), implicit functions, inverse trigonometric functions, exponential and logarithmic functions, and functions given in parametric form. Logarithmic differentiation handles products, quotients and variable exponents. Second-order derivatives are introduced, and the chapter closes with the Mean Value Theorems of Rolle and Lagrange, which connect the derivative to the behaviour of the function on an interval.