Continuity & Differentiability – Complete JEE Main 2026 Revision

This chapter connects limits → graphs → calculus. Most JEE errors happen due to confusing continuity with differentiability.

1. Continuity

A function $f(x)$ is continuous at $x=a$ if and only if:

$\displaystyle \lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=f(a) $

All three conditions must hold simultaneously.

Types of Discontinuity (JEE Favourite)

Type	Description
Removable	LHL = RHL but ≠ f(a)
Jump	LHL ≠ RHL
Infinite	Limit → ±∞
Oscillatory	No definite approach value

📌 Polynomials, exponential, trigonometric functions are continuous everywhere.

2. Differentiability

A function is differentiable at $x=a$ if:

$\displaystyle \lim_{h\to0}\frac{f(a+h)-f(a)}{h} $ exists.

✔ Differentiability ⇒ Continuity ❌ Continuity ⇏ Differentiability

Non-Differentiable Points (Very Important)

• Sharp corners (e.g. $|x|$ at $0$)
• Cusps
• Vertical tangents
• Discontinuity points

3. Modulus Function

$f(x)=|x|$ is:

• Continuous everywhere
• Not differentiable at $x=0$

General rule:

$|x-a|$ is not differentiable at $x=a$

4. Standard Derivatives (Must Memorize)

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

$\frac{d}{dx}(\sin x)=\cos x$

$\frac{d}{dx}(\cos x)=-\sin x$

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

$\frac{d}{dx}(\ln x)=\frac{1}{x}$

5. Chain Rule

If $y=f(g(x))$ then:

$\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$

Extensively used in composite functions.

6. Logarithmic Differentiation

Used when variables appear in both base and power.

$y=x^x \Rightarrow \ln y=x\ln x$

$\frac{1}{y}\frac{dy}{dx}=\ln x+1$

7. Implicit Differentiation

When $y$ is not explicitly defined.

$x^2+y^2=1$

$\frac{dy}{dx}=-\frac{x}{y}$

8. Left & Right Derivatives

For differentiability:

LHD = RHD

Used heavily in piecewise functions.

9. JEE Main Traps & Pain Points

• Assuming continuity implies differentiability
• Forgetting modulus breakdown
• Ignoring one-sided derivatives
• Skipping domain restrictions

10. 10-Second Recall Ladder

• Differentiable ⇒ Continuous
• Not differentiable at sharp points
• Check LHD = RHD
• Modulus breaks slope

✔ Fully aligned with JEE Main 2026 syllabus
✔ Covers all PYQ logic
✔ Foundation for AOD, Tangents & Graphs