CBSE Class 12 Maths Chapter Notes: Continuity And Differentiability

Key Definitions

Limit of a function

A function f(x) has a limit L as x approaches a if f(x) gets arbitrarily close to L as x gets arbitrarily close to a, without necessarily being equal to a. Mathematically, lim (x→a) f(x) = L if lim (x→a-) f(x) = lim (x→a+) f(x) = L (Left Hand Limit = Right Hand Limit).

Continuity at a point

A function f(x) is said to be continuous at a point x = a if lim (x→a) f(x) = f(a). This means three conditions must be met: f(a) is defined, lim (x→a) f(x) exists, and lim (x→a) f(x) = f(a).

Continuity in an interval

A function f(x) is continuous in an open interval (a, b) if it is continuous at every point in the interval. It is continuous in a closed interval [a, b] if it is continuous in (a, b), continuous from the right at a (lim (x→a+) f(x) = f(a)), and continuous from the left at b (lim (x→b-) f(x) = f(b)).

Differentiability at a point

A function f(x) is differentiable at a point x = a if its derivative f'(a) exists. This means lim (h→0) [f(a+h) - f(a)] / h exists. Equivalently, the Left Hand Derivative (LHD) must equal the Right Hand Derivative (RHD) at x = a.

Left Hand Derivative (LHD)

LHD at x = a is lim (h→0-) [f(a+h) - f(a)] / h or lim (x→a-) [f(x) - f(a)] / (x - a).

Right Hand Derivative (RHD)

RHD at x = a is lim (h→0+) [f(a+h) - f(a)] / h or lim (x→a+) [f(x) - f(a)] / (x - a).

Chain Rule

If y = f(u) and u = g(x), then dy/dx = dy/du * du/dx. Used for differentiating composite functions.

Understanding Continuity and Differentiability

At its core, continuity of a function describes a curve that can be drawn without lifting the pen from the paper. Graphically, there are no breaks, jumps, or holes. Formally, a function f(x) is continuous at a point 'a' if the function value at a (f(a)) is defined, the limit of the function as x approaches 'a' exists, and these two values are equal. That is, lim (x→a) f(x) = f(a). This simple equation encapsulates the three necessary conditions for continuity. If any of these conditions fail, the function is discontinuous at that point. Common discontinuities include removable discontinuities (a hole), jump discontinuities (a break where the left and right limits are different), and infinite discontinuities (like asymptotes).

Differentiability, on the other hand, is a stronger condition than continuity. A function is differentiable at a point 'a' if it has a well-defined tangent line at that point. This means the curve must be smooth at 'a', without any sharp corners (like the vertex of a V-shape) or cusps, and also not have any vertical tangent lines. The derivative f'(a) represents the instantaneous rate of change or the slope of the tangent line at x=a. For f(x) to be differentiable at x=a, the Left Hand Derivative (LHD) must be equal to the Right Hand Derivative (RHD) at that point. An important theorem states that if a function is differentiable at a point, it must also be continuous at that point. However, the converse is not true; a function can be continuous but not differentiable (e.g., f(x) = |x| at x=0). This distinction is crucial for exam problems.

Steps to Check Continuity and Differentiability

1. — 1. Find f(a): Evaluate the function at the given point a. If f(a) is undefined, the function is discontinuous. 2. Find LHL: Calculate lim (x→a-) f(x). 3. Find RHL: Calculate lim (x→a+) f(x). 4. Compare: If LHL = RHL = f(a), then the function is continuous at x = a. Otherwise, it is discontinuous.
2. — 1. Check Continuity First: A function must be continuous to be differentiable. If it's discontinuous, it's not differentiable. 2. Calculate LHD: LHD = lim (h→0-) [f(a+h) - f(a)] / h. 3. Calculate RHD: RHD = lim (h→0+) [f(a+h) - f(a)] / h. 4. Compare: If LHD = RHD, then the function is differentiable at x = a, and this common value is f'(a). Otherwise, it is not differentiable.

Must Remember: Key Properties & Theorems

• Every polynomial function is continuous and differentiable everywhere (for all real numbers).
• Exponential functions (e^x, a^x) are continuous and differentiable everywhere.
• Logarithmic functions (log x) are continuous and differentiable in their domain (x > 0).
• Trigonometric functions (sin x, cos x) are continuous and differentiable everywhere. tan x, sec x are continuous and differentiable in their domains.
• The sum, difference, product, and quotient (denominator not zero) of two continuous functions are continuous.
• The sum, difference, product, and quotient (denominator not zero) of two differentiable functions are differentiable.
• If a function f(x) is differentiable at a point x=a, then it is necessarily continuous at x=a. The converse is NOT true.
• A continuous function may not be differentiable (e.g., f(x) = |x| at x=0 has a sharp corner).
• Differentiability implies smoothness. Points of non-differentiability often correspond to sharp turns, cusps, or vertical tangents.

Worked Examples

• {"title":"Example 1: Checking Continuity","bodyMarkdown":"Question: Examine the continuity of the function f(x) = 2x + 3 at x = 1.\nSolution:\n1. f(1) = 2(1) + 3 = 5.\n2. LHL = lim (x→1-) (2x + 3) = 2(1) + 3 = 5.\n3. RHL = lim (x→1+) (2x + 3) = 2(1) + 3 = 5.\nSince LHL = RHL = f(1) = 5, the function f(x) is continuous at x = 1."}
• {"title":"Example 2: Checking Differentiability","bodyMarkdown":"Question: Discuss the differentiability of f(x) = |x| at x = 0.\nSolution:\nFirst, f(x) = |x| is continuous at x=0 because LHL = lim (x→0-) -x = 0, RHL = lim (x→0+) x = 0, and f(0) = 0. So, LHL = RHL = f(0).\nNow, for differentiability:\nLHD = lim (h→0-) [f(0+h) - f(0)] / h = lim (h→0-) [|h| - 0] / h = lim (h→0-) (-h) / h = -1.\nRHD = lim (h→0+) [f(0+h) - f(0)] / h = lim (h→0+) [|h| - 0] / h = lim (h→0+) h / h = 1.\nSince LHD ≠ RHD ( -1 ≠ 1), f(x) = |x| is not differentiable at x = 0."}

Exam Traps & Scoring Tips

Watch out for piecewise functions and modulus functions (|x|, |ax+b|) – these are prime candidates for discontinuity or non-differentiability points, typically at the values where the definition changes or the expression inside the modulus becomes zero. Always check continuity before differentiability. If a function is found to be discontinuous at a point, you can immediately conclude it's not differentiable there, saving time. When using the LHD/RHD formula, be careful with the sign of h as it approaches 0- or 0+. For questions involving composite functions, remember the Chain Rule well. Always show all steps clearly, especially for limit calculations, as step-wise marking is common in CBSE exams.

Practice Questions with Solutions

• Q: Is f(x) = sin(x) continuous and differentiable for all real numbers? A: Yes, sin(x) is both continuous and differentiable for all real numbers.
• Q: If a function g(x) is discontinuous at x=c, can it be differentiable at x=c? A: No, if a function is discontinuous at a point, it cannot be differentiable at that point. Differentiability implies continuity.
• Q: For what values of 'a' and 'b' is the function f(x) = {ax+1, if x <= 3; bx+3, if x > 3} continuous at x=3? A: For continuity at x=3, f(3) = LHL = RHL. So, 3a+1 = 3b+3. This gives 3a - 3b = 2.
• Q: What is the derivative of y = (sin x)^2? A: Using the chain rule, dy/dx = 2(sin x) * (cos x) = sin(2x).

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