CBSE Class 12 Maths Chapter 5 Notes: Continuity and Differentiability

Understanding Continuity and Differentiability: The Foundation of Calculus

Continuity and Differentiability are fundamental concepts in Calculus, serving as the bedrock for understanding how functions behave and enabling the study of rates of change. A continuous function is intuitively understood as one whose graph can be drawn without lifting the pen, implying no breaks, jumps, or holes in its domain. Mathematically, it signifies that a function's value at a point perfectly aligns with its limit as the input variable approaches that point. This smooth, unbroken characteristic is crucial for many real-world applications, from physics to economics.

On the other hand, differentiability refers to the ability to find a unique tangent line at every point on a function's graph. Geometrically, this means the function's graph is 'smooth' and free from sharp corners (cusps) or abrupt changes in direction. The derivative, which arises from differentiability, measures the instantaneous rate of change of a function. A key relationship between these two concepts is that every differentiable function is necessarily continuous, but the converse is not always true – a function can be continuous yet not differentiable (e.g., at sharp corners). This chapter deeply explores these interconnected ideas, their properties, and how to apply them to various classes of functions, including algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions, which are essential tools in higher mathematics and scientific fields.

Key Definitions

Continuity

A function f(x) is continuous at x=a if lim (x→a⁻) f(x) = lim (x→a⁺) f(x) = f(a). The graph can be drawn without lifting the pen.

Differentiability

A function f(x) is differentiable at x=a if the derivative f'(a) exists, meaning the Left Hand Derivative (LHD) equals the Right Hand Derivative (RHD) at x=a. Geometrically, it implies a smooth curve without sharp corners.

Limit of a Function

The value that a function 'approaches' as the input (x) approaches some specific value, without necessarily reaching it.

Left Hand Limit (LHL)

The limit of f(x) as x approaches 'a' from values less than 'a' (i.e., x → a⁻).

Right Hand Limit (RHL)

The limit of f(x) as x approaches 'a' from values greater than 'a' (i.e., x → a⁺).

Left Hand Derivative (LHD)

lim (h→0⁻) [f(a+h) - f(a)] / h. Represents the slope of the tangent approaching from the left.

Right Hand Derivative (RHD)

lim (h→0⁺) [f(a+h) - f(a)] / h. Represents the slope of the tangent approaching from the right.

Chain Rule

A formula to compute the derivative of a composite function. If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Implicit Differentiation

A technique used to differentiate functions where y is not explicitly expressed as a function of x, by differentiating both sides of an equation with respect to x, treating y as a function of x.

Continuity of a Function

Differentiability of a Function

Steps to Check Continuity & Differentiability

1. — 1. Calculate the Left Hand Limit (LHL): lim (x→a⁻) f(x). 2. Calculate the Right Hand Limit (RHL): lim (x→a⁺) f(x). 3. Calculate the function value at the point: f(a). 4. If LHL = RHL = f(a), the function is continuous at x=a. Otherwise, it is discontinuous.
2. — 1. (Optional but recommended) First, check for continuity at x=a. If the function is not continuous, it cannot be differentiable. 2. Calculate the Left Hand Derivative (LHD): lim (h→0⁻) [f(a+h) - f(a)] / h. 3. Calculate the Right Hand Derivative (RHD): lim (h→0⁺) [f(a+h) - f(a)] / h. 4. If LHD = RHD, the function is differentiable at x=a. Otherwise, it is not differentiable.

Worked Examples

• Example 1: Checking Continuity Check the continuity of f(x) = x² + 3 at x=1. LHL: lim (x→1⁻) (x² + 3) = (1)² + 3 = 4 RHL: lim (x→1⁺) (x² + 3) = (1)² + 3 = 4 * f(1): (1)² + 3 = 4 Since LHL = RHL = f(1), f(x) is continuous at x=1.
• Example 2: Checking Differentiability Check the differentiability of f(x) = |x| at x=0. First, f(x)=|x| is continuous at x=0 (LHL=RHL=f(0)=0). LHD: lim (h→0⁻) [|0+h| - |0|] / h = lim (h→0⁻) [-h]/h = -1 (since h<0, |h|=-h) * RHD: lim (h→0⁺) [|0+h| - |0|] / h = lim (h→0⁺) h/h = 1 (since h>0, |h|=h) Since LHD ≠ RHD (-1 ≠ 1), f(x) = |x| is not differentiable at x=0.
• Example 3: Chain Rule Find dy/dx if y = sin(x²). Let u = x², then y = sin(u). dy/du = cos(u) and du/dx = 2x. By Chain Rule, dy/dx = (dy/du) (du/dx) = cos(u) 2x = 2x cos(x²).

Must Remember: Key Points for Exam

• All polynomial, exponential, logarithmic, trigonometric, and inverse trigonometric functions are continuous and differentiable in their respective domains.
• A function continuous on a closed interval [a, b] is bounded and attains its maximum and minimum values on that interval.
• Every differentiable function is continuous, but a continuous function is not necessarily differentiable.
• Discontinuities often occur at points where the function definition changes (for piecewise functions) or where the denominator becomes zero.
• Sharp corners (like in |x|) or vertical tangents indicate non-differentiability.
• Remember the standard derivatives and differentiation rules (Product, Quotient, Chain) perfectly.
• Logarithmic differentiation simplifies finding derivatives of complex products/quotients or functions with variables in both base and exponent.
• Implicit differentiation requires careful application of the chain rule to terms involving y.

Exam Strategy & Traps

When solving continuity and differentiability problems, always explicitly state the conditions you are checking (LHL, RHL, f(a), LHD, RHD). For piecewise functions, the crucial points to examine are where the function's definition changes. These are the most common points of potential discontinuity or non-differentiability. Many students forget to check the function value f(a) alongside the limits for continuity. For differentiability, ensure you've first checked for continuity – a discontinuous function cannot be differentiable. Don't rush through calculations involving modulus functions or the greatest integer function, as they often introduce sharp turns or jumps. Practice numerical problems involving finding unknown constants (k, a, b) that make a function continuous or differentiable over a given interval. Clear, step-by-step presentation of your solution is vital for securing full marks.

Practice Questions with Solutions

• Q: State the fundamental condition for a function f(x) to be continuous at a point x=c. A: The fundamental condition is lim (x→c⁻) f(x) = lim (x→c⁺) f(x) = f(c).
• Q: Give an example of a function that is continuous everywhere but not differentiable at exactly one point. A: f(x) = |x| is continuous everywhere but not differentiable at x=0.
• Q: If y = log(cos x), find dy/dx. A: Using the Chain Rule, dy/dx = (1/cos x) * (-sin x) = -tan x.
• Q: For what kind of functions is logarithmic differentiation most advantageous? A: Logarithmic differentiation is particularly useful for functions involving a variable in the exponent (e.g., x^x) or for complex products and quotients of several functions.

Frequently Asked Questions