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Continuity & Differentiability revision notes

Every idea in the Continuity & Differentiability chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 15 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 concept, and "last" is the latest year one appeared. ★ marks ideas 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 must exist, be finite and agree. At an end of the domain only the inside one-sided limit is needed.
  • Not a discontinuity outside the domain. Continuity is checked only where ff is defined. 1/x1/x is continuous on its domain; for e−∣ln⁡x∣e^{-|\ln x|} on (0,∞)(0,\infty), x=0x=0 is never counted.
  • Not skipping the ends of a closed domain. On [p,q][p,q], ff breaks at pp if f(p+)≠f(p)f(p^+)\ne f(p) and at qq if f(q−)≠f(q)f(q^-)\ne f(q). Such ends count among the points of discontinuity.
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. Find each side with the Limits sheet methods: standard limits, expansions, rationalising, 1∞1^\infty.
xxyyx→0+x\to0^+: 12\frac1{\sqrt2}x→0−x\to0^-: −12-\frac1{\sqrt2}
1−cos⁡xx=2 ∣sin⁡(x/2)∣x\frac{\sqrt{1-\cos x}}x=\frac{\sqrt2\,|\sin(x/2)|}x: the two sides differ, no constant works
  • Most used: sin⁡ax/x→a\sin ax/x\to a, ln⁡(1+ax)/x→a\ln(1+ax)/x\to a, (ax−1)/x→ln⁡a(a^x-1)/x\to\ln a, (1−cos⁡ax)/x2→a2/2(1-\cos ax)/x^2\to a^2/2, (1+ax)1/x→ea(1+ax)^{1/x}\to e^a. A 1∞1^\infty form fgf^g tends to elim⁡g(f−1)e^{\lim g(f-1)}.
  • For x3x^3 or x4x^4 below, expand: sin⁡x≈x−x3/6\sin x\approx x-x^3/6, tan⁡x≈x+x3/3\tan x\approx x+x^3/3, cos⁡x≈1−x2/2+x4/24\cos x\approx1-x^2/2+x^4/24; (1+x)n=1+nx+n(n−1)x2/2+…(1+x)^n=1+nx+n(n-1)x^2/2+\dots up to that power. So tan⁡2x−sin⁡2x≈4x3\tan2x-\sin2x\approx4x^3.
  • Not x2=x\sqrt{x^2}=x. 1−cos⁡x=2 ∣sin⁡(x/2)∣\sqrt{1-\cos x}=\sqrt2\,|\sin(x/2)|, so the side matters: 1−cos⁡x/x→1/2\sqrt{1-\cos x}/x\to1/\sqrt2 as x→0+x\to0^+ but −1/2-1/\sqrt2 as x→0−x\to0^-.
  • Not any constant. If the denominator tends to 0, the numerator must too, or the limit is infinite. (a−bcos⁡2x)/x2(a-b\cos2x)/x^2 is finite only if a=ba=b; then it tends to 2b2b.
  • Quick method. Open [x][x], {x}\{x\} and ∣g∣|g| for the side first. As x→2−x\to2^-: [x]=1[x]=1, and x2−5x+6=(x−2)(x−3)>0x^2-5x+6=(x-2)(x-3)>0, so ∣x2−5x+6∣/(5x−x2−6)=−1|x^2-5x+6|/(5x-x^2-6)=-1.
  • Quick method. Near π/2\pi/2 put x=π/2+hx=\pi/2+h: tan⁡2kx=tan⁡2kh→0\tan2kx=\tan2kh\to0, but tan⁡(2k+1)x=−cot⁡(2k+1)h→∞\tan(2k+1)x=-\cot(2k+1)h\to\infty. So tan⁡4x/tan⁡6x→2/3\tan4x/\tan6x\to2/3.
Continuity of combinations
3× asked, last 2022
f±g, fg, f/g (g(a)≠0), f∘g are continuous if f,g aref\pm g,\ fg,\ f/g\ (g(a)\ne0),\ f\circ g\ \text{are continuous if } f,g \text{ are}
f∘gf\circ g is continuous at aa if gg is continuous at aa and ff at g(a)g(a). Continuous + discontinuous is always discontinuous.
  • Not only the breaks of ff. g∘fg\circ f can also break where f(x)f(x) crosses a break of gg. If gg jumps at 0 and f(x)=x+1f(x)=x+1 for x<0x<0, then g∘fg\circ f jumps at x=−1x=-1.

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Practise Continuity & Differentiability

12 more concepts 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 irremovable discontinuity2×
  • Greatest integer [f(x)]10×
  • Modulus, max and min keep continuity4×

Differentiability

  • Left and right derivatives8×
  • Differentiable implies continuous5×
  • Constants that make f differentiable10×

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

  • Functions defined by an integral3×

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