Class 12 Maths Continuity and Differentiability Notes

5.1 Introduction

Calculus is the study of change and motion, and derivatives are at its core. To study derivatives, we first need the concepts of continuity and differentiability.

• Continuity tells us whether a function is smooth or has jumps/discontinuities.
• Differentiability tells us whether the function has a well-defined slope at a point.

This chapter forms the foundation for differentiation in various forms used in calculus.

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5.2 Continuity

A function f(x) is said to be continuous at a point x = a if the following conditions are satisfied:

1. f(a) is defined
2. Limit of f(x) as x → a exists
3. Limit equals the function value:

$$\lim_{x \to a} f(x) = f(a)$$x→alim​f(x)=f(a)

🔹 Types of Discontinuity

1. Removable Discontinuity: Can be removed by redefining f(a).
2. Jump Discontinuity: The left and right limits exist but are unequal.
3. Infinite Discontinuity: Function approaches infinity near the point.

A function continuous at every point of its domain is called a continuous function.

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5.3 Differentiability

A function f(x) is differentiable at x = a if its derivative exists at that point:$$f'(a) = \lim_{h \to 0} \frac{f(a+h) – f(a)}{h}$$f′(a)=h→0lim​hf(a+h)−f(a)​

• Differentiability implies continuity, but the reverse is not always true.
• A function can be continuous but not differentiable (example: |x| at x = 0).

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5.4 Exponential and Logarithmic Functions

🔹 Exponential Function

• f(x) = e^x
• Derivative: $\frac{d}{dx} e^x = e^x$dxd​ex=ex

🔹 Logarithmic Function

• f(x) = ln x
• Derivative: $\frac{d}{dx} \ln x = \frac{1}{x}$dxd​lnx=x1​

These functions are widely used in growth, decay, and calculus problems.

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5.5 Logarithmic Differentiation

Used when functions are products, quotients, or powers. Steps:

1. Take natural logarithm of both sides: $\ln y = \ln f(x)$lny=lnf(x)
2. Differentiate implicitly using the chain rule
3. Solve for $\frac{dy}{dx}$dxdy​

Simplifies differentiation of complicated expressions like $y = x^x$y=xx or $y = (\sin x)^{\tan x}$y=(sinx)tanx.

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5.6 Derivatives of Functions in Parametric Forms

Sometimes, x and y are expressed as functions of a parameter t:$$x = \phi(t), \quad y = \psi(t)$$x=ϕ(t),y=ψ(t)

• Derivative of y w.r.t x:

$$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{\psi'(t)}{\phi'(t)}$$dxdy​=dx/dtdy/dt​=ϕ′(t)ψ′(t)​

• Second derivative:

$$\frac{d^2y}{dx^2} = \frac{d}{dx} \left( \frac{dy}{dx} \right) = \frac{\frac{d}{dt} \left( \frac{dy}{dx} \right)}{dx/dt}$$dx2d2y​=dxd​(dxdy​)=dx/dtdtd​(dxdy​)​

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5.7 Second Order Derivative

The second derivative measures the rate of change of the first derivative.$$f”(x) = \frac{d^2y}{dx^2} = \frac{d}{dx} \left( f'(x) \right)$$f′′(x)=dx2d2y​=dxd​(f′(x))

• Indicates concavity of the function:
  • $f”(x) > 0$f′′(x)>0: Concave upward
  • $f”(x) < 0$f′′(x)<0: Concave downward
• Helps in finding maxima and minima of functions.

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✅ Key Points to Remember

• Continuity is necessary for differentiability.
• Differentiability implies smoothness; non-differentiable points may have corners or cusps.
• Exponential and logarithmic functions have unique derivative rules.
• Parametric derivatives and second order derivatives are important in mechanics and motion problems.