PYQ (JEE Main & Advanced)

JEE Main Concepts for Continuity & Differentiability

1. Points of Non-Differentiability (Sharp Corners)

Functions involving modulus, like $f(x) = |x-a|$, are continuous everywhere but not differentiable at $x=a$ because the left-hand derivative and right-hand derivative are unequal there. If several modulus terms are added, such as $|x-1|+|x-2|$, the candidate points of non-differentiability are the points where each modulus changes sign.

2. Piecewise Functions and Boundary Points

For differentiability at a boundary point $x=c$, the function must first be continuous there. So in such problems:

1. First impose $\lim_{x \to c^-} f(x)=\lim_{x \to c^+} f(x)=f(c)$.
2. Then impose $LHD=RHD$. If continuity is skipped, one condition is missed and the answer becomes incomplete.

3. Max/Min Functions

For $f(x)=\max(g(x),h(x))$ or $f(x)=\min(g(x),h(x))$, the important points are usually the intersection points of $g(x)$ and $h(x)$. At each such point, one must check whether the left derivative and right derivative match.

4. Infinite Nested Expressions

If $$y=\sqrt{f(x)+\sqrt{f(x)+\cdots}},$$ then by self-similarity, $$y=\sqrt{f(x)+y}.$$ Squaring converts the problem into an implicit relation that can be differentiated easily.

5. Second Derivative of Parametric Functions

If $x=f(t)$ and $y=g(t)$, then $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}.$$ But $$\frac{d^2y}{dx^2}\neq \frac{d^2y/dt^2}{d^2x/dt^2}.$$ The correct formula is $$\frac{d^2y}{dx^2}=\frac{d}{dt}\left(\frac{dy}{dx}\right)\cdot\frac{dt}{dx}.$$

6. Functional Equations and Derivatives

In problems involving equations such as $f(x+y)=f(x)f(y)$, use the first principle or substitute $x+h$ strategically. Such questions often reduce to a differential equation or a constant ratio.

7. Oscillatory Functions near the Origin

For functions like $x^p\sin(1/x)$ or $x^p\cos(1/x)$, continuity and differentiability at $x=0$ depend on the power of $x$. The oscillatory term stays bounded, so the power of $x$ controls the limit.

8. Composite Functions

For repeated compositions like $f(f(f(x)))$, apply the chain rule repeatedly: $$\frac{d}{dx}f(f(f(x)))=f'(f(f(x)))\cdot f'(f(x))\cdot f'(x).$$

9. Logarithmic Differentiation

Whenever both base and exponent are variable, such as $x^{\sin x}$, $(\sin x)^{\cos x}$, or $x^{\ln x}$, logarithmic differentiation is the natural tool.

10. Geometry of Non-Differentiability

A function can fail to be differentiable due to a sharp corner, cusp, vertical tangent, or oscillatory behavior. In JEE Main, sharp-corner and modulus-based questions are especially frequent.