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"Is the discontinuity removable?" is a short question with a precise answer, yet students often decide by looking at a sketch. The sketch can mislead, because a gap in the graph is not always a discontinuity. This post gives a three-number test that settles the type every time, works five examples and points out where NCERT's definition differs from casual usage.
Where is this in the syllabus?
The JEE (Main) 2026 syllabus lists "limits, continuity and differentiability" in Unit 7. The JEE (Advanced) 2026 brochure lists continuity of a function, continuity of composite functions and the intermediate value property. The NCERT Class 12 chapter Continuity and Differentiability (reprint 2026-27) defines continuity and works piecewise examples, but it does not use the words "removable" or "jump". Those labels come from JEE practice; the test below uses only NCERT's definition. Check the current year's documents.
What does NCERT's definition require?
NCERT's Definition 1 applies to a point c in the domain of f: f is continuous at c if the limit of f(x) as x → c equals f(c). In practice that means three numbers must exist and be equal:
the left-hand limit, LHL
the right-hand limit, RHL
the value f(c)
If f is not continuous at c, NCERT calls c a point of discontinuity.
The words "in the domain" matter. NCERT's Example 12 has f(x) = x + 2 for x < 0 and −x + 2 for x > 0, with nothing defined at 0. The graph has a gap, yet NCERT concludes that f is continuous, because 0 is not in its domain. A gap at a point where f is not defined is not, by this definition, a discontinuity.
How do you tell the types apart?
Once c is in the domain and the function is discontinuous there:
Removable: LHL and RHL are finite and equal, but differ from f(c). Changing the single value f(c) to the common limit makes f continuous.
Jump: LHL and RHL are both finite but unequal. The size of the jump is |RHL − LHL|. No choice of f(c) can fix it.
Neither: at least one one-sided limit is infinite or does not exist at all.
Worked example 1: a removable discontinuity
f(x) = (x² − 9)/(x − 3) for x ≠ 3, and f(3) = 5.
For x ≠ 3, (x² − 9)/(x − 3) = x + 3.
LHL = RHL = 3 + 3 = 6.
f(3) = 5, which is not 6.
The limits agree with each other but not with f(3). Redefine f(3) = 6 and the function becomes continuous. Removable.
Worked example 2: a jump
f(x) = 2x + 1 for x < 1, and x² + 4 for x ≥ 1.
LHL = 2(1) + 1 = 3.
RHL = 1 + 4 = 5, and f(1) = 5.
The function is continuous from the right but not from the left. The jump is 5 − 3 = 2, and no value of f(1) can equal both 3 and 5. NCERT's Examples 10 and 11 have the same structure, with one-sided limits of 3 and −1 at x = 1.
Worked example 3: the greatest integer function
NCERT's Example 15 shows that f(x) = [x] is discontinuous at every integer. At an integer c, values just to the left are c − 1 and values just to the right are c, so LHL = c − 1 and RHL = c = f(c). Every integer is a jump of size 1. At non-integers the function is continuous.
Worked example 4: choose k
f(x) = (√(1 + x) − 1)/x for x ≠ 0, and f(0) = k. Find k so that f is continuous at 0.
Multiply top and bottom by √(1 + x) + 1: the expression becomes x ÷ [x(√(1 + x) + 1)] = 1/(√(1 + x) + 1) for x ≠ 0.
As x → 0 from either side, this tends to 1/2.
So k = 1/2.
Check. At x = 0.0001 the original expression is about 0.499988, and at x = −0.0001 about 0.500013.
Any "find k for continuity" question is a removable discontinuity in disguise: the limit exists, and you are choosing f(c) to match it.
Worked example 5: a sign function
f(x) = |x − 2|/(x − 2) for x ≠ 2, and f(2) = 1. For x > 2 the value is 1; for x < 2 it is −1. LHL = −1 and RHL = 1, so there is a jump of 2 at x = 2. If f(2) had not been defined at all, 2 would not be in the domain, and as in NCERT's Example 12, f would be continuous on its domain.
Which mistakes cost marks?
Calling a point outside the domain a discontinuity, against NCERT's definition.
Computing only one side. Piecewise functions, |x| and [x] always need both.
Calling a jump removable because f(c) equals one of the one-sided limits.
Forgetting to simplify before taking a 0/0 limit, as in Example 4.
Judging from a sketch instead of the three numbers.
For the limits themselves, including standard limits and indeterminate forms, see our post on limits and continuity for JEE.
If the idea of a limit still feels shaky underneath, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, each with its own lesson and its place in the map.
A practice routine
For each NCERT example in section 5.2, write LHL, RHL and f(c) in one line, then label the type.
Build two removable and two jump examples yourself, then swap with a friend to classify.
Do five "find k" questions and check each answer numerically just either side of the point.
Once a week, classify the discontinuities of [x], x − [x] and [2x] on the interval 0 to 2.
Today, take f(x) = (x³ − 8)/(x − 2) with f(2) = 10, decide the type and fix it if you can.
