JEE Advanced Continuity and Differentiability — previous year questions

35 questions from 2026, 2024, 2023, 2021, 2020, 2019, 2018, 2017, 2016, 2015, 2014, 2012, 2011, 2009, 2008, 2007.

• Consider the function f : (-π/2, π/2) → (-∞, ∞) defined by f(x) = (|x| + |x - 1|) sin x + [x sin x], where [x…2026JEE Advanced
• For a real number α, let [α] denote the greatest integer less than or equal to α. For a finite set S, let |S|…2026JEE Advanced
• Let R denote the set of all real numbers. Let f : R → R be an arbitrary function and let g : R → R be the…2026JEE Advanced
• Let f:R→R and g:R→R be functions defined by f(x)= x|x|sin(1/x), x≠ 0 0, x=0 and g(x)= 1-2x, 0≤ x≤ 1/2, 0,…2024JEE Advanced
• Let R denote the set of all real numbers. Consider the polynomial function f:R→R defined by…2024JEE Advanced
• Let f : (0, 1) → R be the function defined as f(x) = [4x](x - 1/4)²(x - 1/2), where [x] denotes the greatest…2023JEE Advanced
• Let S be the set of all twice differentiable functions f from R to R such that d²f/dx²(x)>0 for all x∈(-1,1).…2023JEE Advanced
• Let S = (0, 1) ∪ (1, 2) ∪ (3, 4) and T = {0, 1, 2, 3}. Then which of the following statements is(are) true?2021JEE Advanced
• Let the functions f:(-1,1)→R and g:(-1,1)→(-1,1) be defined by f(x)=|2x-1|+|2x+1| and g(x)=x-[x], where [x]…2020JEE Advanced
• Let the function f:R→R be defined by f(x)=x³-x²+(x-1)sin x and let g:R→R be an arbitrary function. Let fg:R→R…2020JEE Advanced
• Let f: R → R given by f(x)= x⁵+5x⁴+10x³+10x²+3x+1, x<0; x²-x+1, 0 ≤ x<1; 2/3x³-4x²+7x-8/3, 1 ≤ x<3;…2019JEE Advanced
• For a polynomial g(x) with real coefficients, let m_g denote the number of distinct real roots of g(x).…2019JEE Advanced
• Let f:R→R and g:R→R be two non-constant differentiable functions. If f'(x)=(e⁽ᶠ⁽ˣ⁾⁻ᵍ⁽ˣ⁾⁾)g'(x) for all x∈R,…2018JEE Advanced
• Let f₁ : R → R, f₂ : (-π/2, π/2) → R, f₃ : (-1, e^(π/2) - 2) → R and f₄ : R → R be defined by \ (i) f₁(x) =…2018JEE Advanced
• For any positive integer n, define fₙ : (0, ∞) → R as fₙ(x) = ∑ⱼ₌₁ⁿ tan⁻¹(1/(1+(x+j)(x+j-1))) for all x ∈ (0,…2018JEE Advanced
• For every twice differentiable function f:R→[-2,2] with (f(0))²+(f'(0))²=85, which of the following…2018JEE Advanced
• Let f:(0, π) → R be a twice differentiable function such that lim_(t → x) (f(x)sin t - f(t)sin x)/(t - x) =…2018JEE Advanced
• Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the…2017JEE Advanced
• Let f: [-1/2, 2] → R and g: [-1/2, 2] → R be functions defined by f(x) = [x²-3] and g(x) = |x| f(x) + |4x-7|…2016JEE Advanced
• Let a, b ∈ R and f : R → R be defined by f(x) = a cos(|x³ - x|) + b |x| sin(|x³ + x|). Then f is .2016JEE Advanced
• Let f:R→R, g:R→R and h:R→R be differentiable functions such that f(x)=x³+3x+2, g(f(x))=x and h(g(g(x)))=x for…2016JEE Advanced
• Let g : R → R be a differentiable function with g(0) = 0, g'(0) = 0 and g'(1) ≠ 0. Let f(x) = x/|x| g(x), x ≠…2015JEE Advanced
• For every pair of continuous functions f, g : [0, 1] → R such that max {f(x) : x ∈ [0,1]} = max {g(x) : x ∈…2014JEE Advanced
• Let f: R -> R and g: R -> R be respectively given by f(x) = |x| + 1 and g(x) = x^2 + 1. Define h: R -> R by…2014JEE Advanced
• Let f₁ : R → R, f₂ : [0, ∞) → R, f₃ : R → R and f₄ : R → [0, ∞) be defined by f₁(x) = |x| if x < 0, eˣ if x ≥…2014JEE Advanced
• For every integer n, let aₙ and bₙ be real numbers. Let function f: R → R be given by f(x)= aₙ+sin π x, for x…2012JEE Advanced
• Let f:[1,∞)→[2,∞) be a differentiable function such that f(1)=2. If 6∫₁ˣf(t) dt=3xf(x)-x³ for all x≥ 1, then…2011JEE Advanced
• If f(x) = -x - π/2, x ≤ -π/2 -cos x, -π/2 < x ≤ 0 x - 1, 0 < x ≤ 1 ln x, x > 1 , then2011JEE Advanced
• Let f: R → R be a function such that f(x+y)=f(x)+f(y), ∀ x, y ∈ R. If f(x) is differentiable at x=0, then2011JEE Advanced
• Let f(θ) = sin(tan⁻¹(sinθ/(√(cos 2θ)))), where -π/4 < θ < π/4. Then the value of d/d(tanθ)(f(θ)) is ______.2011JEE Advanced
• If the function f(x)=x³+e^(x/2) and g(x)=f⁻¹(x), then the value of g'(1) is ______.2009JEE Advanced
• Let g(x) = log f(x) where f(x) is twice differentiable positive function on (0, ∞) such that f(x+1) = x f(x).…2008JEE Advanced
• Let f and g be real valued functions defined on interval (-1, 1) such that g''(x) is continuous, g(0) ≠ 0,…2008JEE Advanced
• Let f(x) = 2 + cos x for all real x. \ STATEMENT I : For each real t, there exists a point c in [t, t + π]…2007JEE Advanced
• d²x/dy² equals2007JEE Advanced

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