📘 Chapter: Continuity and Differentiability (Class 12 – NCERT Maths)
🔹 1. Introduction
Continuity and Differentiability are fundamental concepts of calculus. They help in studying the behaviour of functions, slopes of curves, and rates of change. This chapter forms the base for applications of derivatives.
🔹 2. Continuity of a Function
A function f(x) is said to be continuous at x = a if all the following three conditions are satisfied:
1️⃣ f(a) is defined
2️⃣ limx→a f(x) exists
3️⃣ limx→a f(x) = f(a)
📌 If any one of the above conditions fails, the function is discontinuous at that point.
Types of Discontinuity
✔ Removable discontinuity
✔ Jump discontinuity
✔ Infinite discontinuity
🔹 3. Continuity in an Interval
A function is continuous:
✔ In an open interval (a, b) if it is continuous at every point in that interval
✔ In a closed interval [a, b] if it is continuous in (a, b), right continuous at a, and left continuous at b
🔹 4. Differentiability
A function f(x) is said to be differentiable at x = a if:
limh→0 [ f(a + h) − f(a) ] / h exists
📌 Differentiability implies continuity, but continuity does NOT always imply differentiability.
🔹 5. Relationship Between Continuity and Differentiability
✔ If a function is differentiable at a point, it is continuous at that point.
✔ A function may be continuous but not differentiable.
📌 Examples where function is not differentiable:
✔ Sharp corners
✔ Cusps
✔ Vertical tangents
🔹 6. Derivatives of Basic Functions
Some standard derivatives:
d/dx (xn) = n xn−1
d/dx (sin x) = cos x
d/dx (cos x) = −sin x
d/dx (ex) = ex
d/dx (log x) = 1/x
🔹 7. Derivatives of Trigonometric Functions
d/dx (tan x) = sec2 x
d/dx (cot x) = −cosec2 x
d/dx (sec x) = sec x tan x
d/dx (cosec x) = −cosec x cot x
🔹 8. Derivatives of Inverse Trigonometric Functions
d/dx (sin−1 x) = 1 / √(1 − x²)
d/dx (cos−1 x) = −1 / √(1 − x²)
d/dx (tan−1 x) = 1 / (1 + x²)
🔹 9. Derivatives of Implicit Functions
When y is not explicitly expressed in terms of x, differentiation is done with respect to x on both sides.
📌 Remember to differentiate y as dy/dx.
🔹 10. Logarithmic Differentiation
Used when functions are complicated, especially involving powers, products, or quotients.
Steps:
1️⃣ Take log on both sides
2️⃣ Differentiate implicitly
3️⃣ Simplify the result
🔹 11. Derivatives of Parametric Functions
If x and y are given in terms of a parameter t:
dy/dx = (dy/dt) / (dx/dt)
🔹 12. Second Order Derivatives
The derivative of dy/dx with respect to x is called the second order derivative.
It is denoted by:
d²y/dx²
⭐ Key Takeaways
✔ Continuity checks smoothness of function
✔ Differentiability checks existence of slope
✔ Differentiability implies continuity
✔ This chapter is the foundation for applications of derivatives