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Continuity & Differentiability formula sheet

Every formula in the Continuity & Differentiability chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 20 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that formula, and "last" is the latest year one appeared. ★ marks formulas asked 8 or more times.

Continuity at a point

Continuity at a point
4× asked, last 2024
lim⁡x→a−f(x)=lim⁡x→a+f(x)=f(a)\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=f(a)
All three exist, finite, equal. Only points of the domain count; at an end of [p,q][p,q] use the inside limit.
Constants that make f continuous
26× asked, last 2025
lim⁡x→a−f1(x)=f(a)=lim⁡x→a+f2(x)\lim_{x\to a^-}f_1(x)=f(a)=\lim_{x\to a^+}f_2(x)
One equation per side. Denominator →0\to0 forces numerator →0\to0: (a−bcos⁡2x)/x2(a-b\cos2x)/x^2 needs a=ba=b, limit 2b2b.
Limits used most
sin⁡axx→a,ax−1x→ln⁡a,1−cos⁡axx2→a22fg (1∞)→elim⁡g(f−1)\begin{aligned}&\frac{\sin ax}x\to a,\quad \frac{a^x-1}x\to\ln a,\quad \frac{1-\cos ax}{x^2}\to\frac{a^2}2\\&f^g\ (1^\infty)\to e^{\lim g(f-1)}\end{aligned}
sin⁡x≈x−x3/6\sin x\approx x-x^3/6, tan⁡x≈x+x3/3\tan x\approx x+x^3/3, so tan⁡2x−sin⁡2x≈4x3\tan2x-\sin2x\approx4x^3. 1−cos⁡x=2 ∣sin⁡(x/2)∣\sqrt{1-\cos x}=\sqrt2\,|\sin(x/2)|: sides differ.
Continuity of combinations
3× asked, last 2022
f±g, fg, f/g (g(a)≠0), f∘g: continuousf\pm g,\ fg,\ f/g\ (g(a)\ne0),\ f\circ g:\ \text{continuous}
Continuous + discontinuous is discontinuous. g∘fg\circ f also breaks where f(x)f(x) hits a break of gg.

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Practise Continuity & Differentiability

16 more formulas in 4 topics

Types of discontinuity; [x], |x|, max and min; Differentiability; Counting points of non-differentiability; and Functions defined by a limit or an integral. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Types of discontinuity; [x], |x|, max and min

  • Removable and jump2×
  • Greatest integer [f(x)]10×
  • Fractional part and counting
  • Max and min keep continuity4×

Differentiability

  • Left and right derivatives8×
  • Differentiable implies continuous5×
  • xⁿ sin(1/x)
  • Constants that make f differentiable10×
  • Lipschitz-type bound

Counting points of non-differentiability

  • Corners of |f(x)|7×
  • |x − a| times g(x)3×
  • Max and min of two curves10×
  • Differentiability of combinations3×
  • f(|x|) at x = 0

Functions defined by a limit or an integral

  • Defined by an integral3×
  • Defined by a limit

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Practise Continuity & Differentiability

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Version 1, October 2026. Free to share for study, as it is. Please don't sell it or remove the PrepWiser name.

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