Chapter 5: Continuity and Differentiability

Summary

• Continuation of differentiation study from Class XI.
• Introduction of continuity and differentiability concepts.
• Differentiation of inverse trigonometric functions.
• Introduction of exponential and logarithmic functions.
• Fundamental theorems in differential calculus.

Key Concepts

Continuity

• A function is continuous at a point if the limit at that point equals the function's value.
• Functions can be continuous on their entire domain.
• Sum, difference, product, and quotient of continuous functions are also continuous.
• Every differentiable function is continuous, but not vice versa.

Differentiability

• The derivative of a function at a point is defined using limits:
  $f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}$

Exponential and Logarithmic Functions

• Exponential functions grow rapidly; logarithmic functions are their inverses.
• Logarithmic differentiation is used for functions of the form $f(x) = [u(x)]^{v(x)}$.

Important Theorems

• If $f$ and $g$ are continuous at $c$, then:
  • $f + g$ is continuous at $c$
  • $f - g$ is continuous at $c$
  • $f \cdot g$ is continuous at $c$
  • $\frac{f}{g}$ is continuous at $c$ if $g(c) \neq 0$

Exercises

1. Prove that the function $f(x) = 5x - 3$ is continuous at specified points.
2. Examine the continuity of various functions at given points.
3. Differentiate functions using the chain rule and logarithmic differentiation.

Examples

• Example of checking continuity at specific points.
• Example of differentiating functions using various methods.

Chapter 5: Continuity and Differentiability

5.1 Introduction

• Continuation of differentiation study from Class XI.
• Concepts introduced:
  • Continuity
  • Differentiability
  • Inverse trigonometric functions
  • Exponential and logarithmic functions
• Fundamental theorems in differential calculus.

5.2 Continuity

• Definition: A function is continuous at a point if the limit at that point equals the function's value.
• Example: Function defined as:
  • f(x) = 1, if x ≤ 0
  • f(x) = 2, if x > 0
• Graph Analysis: At x = 0, the function is not continuous.

Exercises

1. Prove that f(x) = 5x - 3 is continuous at x = 0, x = -3, and x = 5.
2. Examine continuity of f(x) = 2x² - 1 at x = 3.
3. Examine the following functions for continuity:
   • (a) f(x) = 5
   • (b) f(x) = x
   • (c) f(x) = |x - 5|

5.3 Differentiability

• Definition: The derivative of a real function at a point c is defined as:
  • lim (f(c+h) - f(c)) / h as h approaches 0.
• Key Point: Every differentiable function is continuous, but not vice versa.

5.4 Exponential and Logarithmic Functions

• Introduction to exponential functions and logarithmic functions.
• Natural Exponential Function: y = e^x.
• Logarithm Definition: For b > 1, log_b(a) = x if b^x = a.

Important Observations

• Logarithm function is defined from positive real numbers to all real numbers.
• Common logarithm (base 10) and natural logarithm (base e).

5.5 Logarithmic Differentiation

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

5.6 Derivatives of Functions in Parametric Forms

• Relationships between variables expressed via a third variable.
• Example: x = a cos³(θ), y = a sin³(θ).

5.7 Miscellaneous Exercises

• Differentiate various functions and find second-order derivatives.

Common Mistakes and Exam Tips

Common Pitfalls

• Misunderstanding Continuity: Students often confuse the conditions for a function to be continuous. Remember that a function must be defined at the point, and the limit must equal the function's value at that point.
• Ignoring Piecewise Functions: When dealing with piecewise functions, ensure to check continuity at the boundaries where the definition changes.
• Limit Calculation Errors: Be careful with limit calculations, especially at points of discontinuity. Ensure to evaluate both left-hand and right-hand limits.
• Differentiability vs. Continuity: A common mistake is assuming that if a function is continuous, it is also differentiable. Remember that a function can be continuous but not differentiable at certain points.

Tips for Success

• Practice with Examples: Work through various examples of continuity and differentiability to solidify understanding. Pay attention to the definitions and properties of functions.
• Graph Functions: Sketching graphs can help visualize continuity and differentiability. Look for breaks, jumps, or corners in the graph that indicate discontinuity or non-differentiability.
• Review Definitions: Regularly review the definitions of continuity and differentiability, and practice applying them to different types of functions.
• Use Theorems: Familiarize yourself with fundamental theorems related to continuity and differentiability, as they can provide shortcuts in problem-solving.