Continuity and Differentiability

Introduction

• Continuation of differentiation study from Class XI.
• Introduction to concepts of continuity, differentiability, and their relationships.
• Differentiation of inverse trigonometric functions.
• Introduction of exponential and logarithmic functions.
• Exploration of fundamental theorems in calculus.

Continuity

• Informal Example of Continuity: Function values close on x-axis except at x = 0.

• Formal Definition: A function f is continuous at point c if

  [\lim_{{x \to c}} f(x) = f(c)]

  • Left-hand limit, right-hand limit, and value of the function must equal.

• Discontinuity: If a function is not continuous at a point, it is discontinuous there._

Examples of Continuity

• Example 1: Function f(x) = 2x + 3 is continuous at x = 1.
• Example 2: Function f(x) = x^2 is continuous at x = 0.
• Example 3: Modulus function f(x) = |x| is continuous at x = 0.
• Example 4: Function f(x) with different expressions for x = 0 is not continuous at x = 0.
• Example 5: Constant function f(x) = k is continuous everywhere.

Algebra of Continuous Functions

• Sum, difference, product, and quotient (if denominator not zero) of continuous functions are continuous.
• Rational functions (quotients of polynomials) are continuous at points where the denominator is not zero.

Differentiability

• Derivative Definition: The derivative of f at c is

  [f'(c) = \lim_{{h \to 0}} \frac{{f(c+h) - f(c)}}{h}]

  • If this limit exists, f is differentiable at c.

• Theorem: If f is differentiable at c, it is also continuous at c.

• Differentiability implies continuity but not vice versa (e.g., f(x) = |x| is continuous but not differentiable at 0)._

Chain Rule

• Used for differentiating compositions of functions.

• If f = v 0 u, then

  [\frac{df}{dx} = \frac{dv}{du} \cdot \frac{du}{dx}]

Derivatives of Implicit and Inverse Functions

• Techniques for differentiating implicit functions without solving for y in terms of x.
• Derivatives of inverse trigonometric functions using chain rule.

Exponential and Logarithmic Functions

• Exponential Functions: Functions of the form y = b^x.
  • Grow faster than any polynomial function.
  • y = e^x is the natural exponential function with base e.
• Logarithmic Functions: Inverse of exponential functions.
  • Defined only for positive numbers.
  • log_b a = x if b^x = a.

Logarithmic Differentiation

• Useful technique for differentiating functions of the form [u(x)]^v(x).
• Both f(x) and u(x) must be positive.

Derivatives in Parametric Form

• When x and y are expressed as functions of a parameter t.
• Use chain rule to find (\frac{dy}{dx}) by differentiating (\frac{dy}{dt}) and (\frac{dx}{dt}).

Second Order Derivative

• The second derivative (\frac{d^2y}{dx^2}) measures the rate of change of the rate of change.
• Higher order derivatives can be defined similarly.

Summary

• Continuous functions have limits that equal the function value at each point.
• Differentiability at a point implies continuity at that point.
• Techniques for differentiating products, quotients, implicit functions, and using chain rule are essential for advanced calculus.