Continuity and Differentiability

Notes on Continuity and Differentiability

1. Introduction

The chapter begins by stating the importance of continuity and differentiability in mathematical functions, continuing from previous studies of differentiation in Class XI. These concepts form the basis for further explores new functions, including exponential and logarithmic functions.

2. Continuity

Continuity is defined as follows: A function is continuous at a point c if the limit of the function as it approaches c equals the function's value at that point:

[ \lim_{x \to c} f(x) = f(c) ]
This definition implies that for a function to be continuous, three conditions must be met:

• The function must be defined at c,
• The left-hand limit must equal the right-hand limit, and
• The left-hand limit, right-hand limit, and the function value at c must all be equal.

Examples of Continuity:

• Example 1: The function ( f(x) = 2x + 3 ) is continuous at ( x = 1 ) because ( \lim_{x \to 1} f(x) = 5 = f(1) ).
• Example 2: The function ( f(x) = x^2 ) is continuous at ( x = 0 ) since all limits and values coincide.

3. Points of Discontinuity

A function is discontinuous at a point c if it does not meet the conditions for continuity. The points of discontinuity can be identified through various functions, including piecewise definitions, where limits differ from defined values.

4. Algebra of Continuous Functions

Certain operations yield continuous functions if the original functions involved are continuous:

1. The sum of two continuous functions is continuous.
2. The difference of two continuous functions is continuous.
3. The product of two continuous functions is continuous.
4. The quotient of two continuous functions is continuous, except where the denominator is zero.

5. Differentiability

Differentiability concerns the existence of a derivative at a point. A function is differentiable at point c if: [ f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h} ]
This definition states that if the derivative exists, it implies that the function is continuous at c. However, a continuous function isn't necessarily differentiable.

Important Derivatives:

• Standard Derivatives include:
  • ( \frac{d}{dx} (\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}} )
  • ( \frac{d}{dx} (\cos^{-1} x) = -\frac{1}{\sqrt{1 - x^2}} )
  • ( \frac{d}{dx} (\tan^{-1} x) = \frac{1}{1 + x^2} )
  • ( \frac{d}{dx} (e^x) = e^x )
• Chain Rule: For composite functions, if ( f = v(u(x)) ), then: [ f'(x) = v'(u) \cdot u'(x) ]

6. Inverse Trigonometric Functions Differentiation

The derivatives of inverse trigonometric functions can also be computed by the chain rule, taking care to apply the appropriate limits:

• The derivatives include:
  • ( \frac{d}{dx} (\tan^{-1} x) = \frac{1}{1 + x^2} )
  • ( \frac{d}{dx} (\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}} )

7. Exponential and Logarithmic Functions

Exponential functions of the form ( f(x) = a^x ) grow rapidly, surpassing polynomial functions as x increases. The logarithm serves as the inverse function to exponentiation, having properties such as:

• Can only operate over positive real numbers.
• ( \log(ab) = \log(a) + \log(b) )

8. Logarithmic Differentiation

Logarithmic differentiation is useful for differentiating products and power functions. Taking the logarithm of both sides can simplify complex derivatives.

Conclusion

Understanding these concepts establishes a foundation for further studies in calculus and applied mathematics. By grasping the linked nature of continuity and differentiability, students strengthen their analytical skills in studying function behavior in various contexts.

1. A function is continuous at a point if ( \lim f(x) = f(c) ).
2. Points of discontinuity exist when limits do not comply with function values at those points.
3. Continuous functions yield a continuous sum, difference, product, and quotient (except at undefined points).
4. A function must be differentiable at a point to imply it is continuous there.
5. Chain Rule applies for differentiating composite functions.
6. Standard derivatives: ( \frac{d}{dx}(e^x) = e^x, \frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}} ).
7. Exponential functions grow faster than polynomial functions.
8. Logarithmic differentiation is a technique useful in simplifying derivatives of complex functions.
9. The logarithm is only defined for positive real numbers and has a range of all real numbers.
10. The inverse trigonometric functions have specific derivatives that are foundational in further calculus.