📌 1. Introduction

Continuity and differentiability describe how functions behave.

Continuity ensures no breaks or jumps
Differentiability tells us how a function changes (rate of change)

These concepts are widely used in physics, engineering, economics, and data science.

📖 2. Continuity of a Function

A function $f(x)$ f(x) is said to be continuous at a point x=ax = ax=a if:

$f(a)$ f(a) is defined
$\lim_{x \to a} f(x)$ limx→af(x) exists
$\lim_{x \to a} f(x) = f(a)$ limx→af(x)=f(a)

🧠 Key Idea:

A function is continuous if you can draw its graph without lifting your pen.

📊 Types of Discontinuity

1. Removable Discontinuity

Hole in graph (limit exists but value differs)

2. Jump Discontinuity

Left and right limits are different

3. Infinite Discontinuity

Function goes to infinity

📌 Continuity in an Interval

Continuous in $(a, b)$ (a,b) → continuous at every point
Continuous in $[a, b]$ [a,b] → continuous inside + endpoints

📘 3. Algebra of Continuous Functions

If $f$ f and $g$ g are continuous, then:

$f + g$ f+g is continuous
$f – g$ f−g is continuous
$fg$ fg is continuous
$\frac{f}{g}$ gf is continuous (if $g(x) \neq 0$ g(x)=0)

📐 4. Standard Continuous Functions

These are always continuous in their domains:

Polynomial functions
Rational functions
Trigonometric functions
Exponential functions
Logarithmic functions

📌 5. Differentiability

A function is differentiable at x=ax = ax=a if: $\lim_{h \to 0} \frac{f(a+h) – f(a)}{h}$ h→0limhf(a+h)−f(a)

exists.

$\lim_{h \to 0} \frac{f(a+h) – f(a)}{h}$ limh→0hf(a+h)−f(a)

This limit is called the derivative of the function at that point.

📊 Relation Between Continuity and Differentiability

Differentiability ⇒ Continuity
Continuity ≠ Differentiability

👉 Example: $f(x) = |x|$ f(x)=∣x∣ is continuous but not differentiable at $x = 0$ x=0

📈 6. Derivative of a Function

The derivative represents rate of change or slope of tangent. $f'(x) = \frac{dy}{dx}$ f′(x)=dxdy

$f'(x) = \frac{dy}{dx}$ f′(x)=dxdy

📌 Geometrical Meaning

Derivative = slope of tangent line
Positive derivative → increasing function
Negative derivative → decreasing function

📚 7. Derivatives of Standard Functions

1. Power Function

$\frac{d}{dx}(x^n) = nx^{n-1}$ dxd(xn)=nxn−1

2. Constant

$\frac{d}{dx}(c) = 0$ dxd(c)=0

3. Trigonometric Functions

$\frac{d}{dx}(\sin x) = \cos x$ dxd(sinx)=cosx
$\frac{d}{dx}(\cos x) = -\sin x$ dxd(cosx)=−sinx
$\frac{d}{dx}(\tan x) = \sec^2 x$ dxd(tanx)=sec2x

4. Exponential

$\frac{d}{dx}(e^x) = e^x$ dxd(ex)=ex

5. Logarithmic

$\frac{d}{dx}(\log x) = \frac{1}{x}$ dxd(logx)=x1

🔗 8. Rules of Differentiation

➕ 1. Sum Rule

$\frac{d}{dx}(u + v) = u’ + v’$ dxd(u+v)=u′+v′

✖️ 2. Product Rule

$\frac{d}{dx}(uv) = u’v + uv’$ dxd(uv)=u′v+uv′

➗ 3. Quotient Rule

$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{vu’ – uv’}{v^2}$ dxd(vu)=v2vu′−uv′

🔄 4. Chain Rule

If $y = f(g(x))$ y=f(g(x)): $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ dxdy=dudy⋅dxdu

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ dxdy=dudy⋅dxdu

🔁 9. Higher Order Derivatives

Second derivative: $\frac{d^2y}{dx^2}$ dx2d2y

Used to check concavity
Helps in maxima & minima

📌 10. Differentiability of Special Functions

1. Modulus Function

$f(x) = |x|$ f(x)=∣x∣

Not differentiable at $x = 0$ x=0

2. Greatest Integer Function

Not continuous at integers → not differentiable

📊 11. Increasing and Decreasing Functions

If $f'(x) > 0$ f′(x)>0 → increasing
If $f'(x) < 0$ f′(x)<0 → decreasing

📉 12. Tangents and Normals

Equation of Tangent:

$y – y_1 = m(x – x_1)$ y−y1=m(x−x1)

Where $m = \frac{dy}{dx}$ m=dxdy

Equation of Normal:

Slope = $-\frac{1}{m}$ −m1

🧩 13. Applications of Derivatives

Finding slope
Rate of change
Motion problems
Optimization
Economics (profit, cost)

📌 14. Important Theorems

Rolle’s Theorem

If:

Function continuous in $[a,b]$ [a,b]
Differentiable in $(a,b)$ (a,b)
$f(a) = f(b)$ f(a)=f(b)

Then ∃ $c$ c such that: $f'(c) = 0$ f′(c)=0

Lagrange’s Mean Value Theorem

$f'(c) = \frac{f(b) – f(a)}{b – a}$ f′(c)=b−af(b)−f(a)

❗ Common Mistakes

Ignoring domain
Wrong application of chain rule
Confusing continuity with differentiability
Sign errors in derivatives

🧠 Exam Tips

Learn standard derivatives
Practice chain rule well
Solve NCERT examples
Focus on graphs

📚 Practice Questions

Check continuity at a point
Find derivative using definition
Apply product/chain rule
Find tangent equation

🎯 Conclusion

Continuity and Differentiability are fundamental concepts in calculus. They help us understand how functions behave and change. Mastering this chapter is essential for exams like JEE, CUET, NDA, and higher studies.