Are these Continuity and Differentiability notes free?

Yes. The full 3-page Continuity and Differentiability sheet is free inside CapCompass, which runs in your browser with no account needed.

What does the Continuity and Differentiability revision sheet cover?

Continuity at a point, condition 1: f(a) is defined, a real number actually comes out, condition 2: the limit approached with x below a matches the limit approached with x above a, condition 3: that common limit equals the function's own value at a, Three ways continuity fails at x=a, Check continuity of a piecewise function at x=1, Algebra of continuous functions, f and g both continuous at a: their sum, difference and product are too, their quotient is continuous at a as well, needs g(a) not zero, a continuous function of a continuous function stays continuous, Differentiability, Continuous vs differentiable at a point.

Is this for Class 12 CBSE Mathematics?

It is written from the NCERT syllabus, so it serves CBSE and any board that follows NCERT for Mathematics.

🧭 CapCompassfree study planner

Continuity and Differentiability — Class 12 CBSE Mathematics revision notes

A 3-page visual revision sheet for Continuity and Differentiability, written for Class 12 CBSE Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.

Continuity and Differentiability revision notes, page 1 of 3 — A preview of page 1 — all 3 pages open full size in the app.

Continuity and Differentiability in one look

an unbroken curve first, a well-defined tangent second: neither one is automatic

Continuity at a point

condition 1: f(a) is defined, a real number actually comes out f(a)
condition 2: the limit approached with x below a matches the limit approached with x above a LHL = RHL
condition 3: that common limit equals the function's own value at a LHL = RHL = f(a)

Three ways continuity fails at x=a

check the limit and the function value at a

LHL and RHL don't match — no limit at all: the two-sided limit does not exist
limit exists, f(a) undefined — a hole in the domain: there's nothing plotted at x=a to compare against
limit and f(a) disagree — a mismatched value: the curve jumps to a different height right at a

Check continuity of a piecewise function at x=1

Q. f(x) = x^2 for x<=1, and f(x) = 2x-1 for x>1. Is f continuous at x=1?

f(1) = 1^2 = 1, using the x^2 branch since 1<=1
limit from below: as x approaches 1 with x<=1, x^2 approaches 1^2 = 1
limit from above: as x approaches 1 with x>1, 2x-1 approaches 2(1)-1 = 1
both one-sided limits equal 1, so the two-sided limit exists and equals 1
compare: the limit (1) matches f(1) (1)

Answer: f is continuous at x=1

Algebra of continuous functions

f and g both continuous at a: their sum, difference and product are too f+g, f-g, fg
their quotient is continuous at a as well, needs g(a) not zero f/g
a continuous function of a continuous function stays continuous g(f(x))

Differentiability

f is differentiable at a when the limit of [f(a+h)-f(a)]/h, as h approaches 0, exists
that same limit must be identical whether h approaches 0 from positive or negative values
differentiable at a point always forces continuous there: no exception to this direction
converse fails: |x| is continuous at x=0 but not differentiable; one-sided limits give -1 and +1

Continuous vs differentiable at a point

Continuous

just needs LHL = RHL = f(a)
a corner, like |x| at 0, is still continuous

Differentiable

needs the same slope approaching from both sides
always implies continuous; the converse fails at a corner

Page 1 of 3. The full sheet, all 3 pages, opens free in the app.

What this Continuity and Differentiability sheet covers

Continuity at a point
condition 1: f(a) is defined, a real number actually comes out
condition 2: the limit approached with x below a matches the limit approached with x above a
condition 3: that common limit equals the function's own value at a
Three ways continuity fails at x=a
Check continuity of a piecewise function at x=1
Algebra of continuous functions
f and g both continuous at a: their sum, difference and product are too
their quotient is continuous at a as well, needs g(a) not zero
a continuous function of a continuous function stays continuous
Differentiability
Continuous vs differentiable at a point
Continuous
Differentiable
Chain rule
y depends on u, and u depends on x
one more layer chains the same way
Standard derivatives: inverse trig, exponential, log
d/dx of sin inverse x
d/dx of tan inverse x
d/dx of e to the x
d/dx of natural log of x, for x>0
Where each standard derivative is even defined
Logarithmic differentiation of x to the power x
Rolle's theorem
conditions: f continuous on [a,b], differentiable on (a,b), f(a)=f(b); conclusion holds for some c strictly between a and b
Lagrange's mean value theorem
conditions: f continuous on [a,b], differentiable on (a,b), no equal-endpoint condition needed; conclusion holds for some c strictly between a and b
Verify Rolle's theorem
Reading the chapter together
Traps that cost marks
Tricks that save time

Open the full Continuity and Differentiability sheet →Free, works offline, no account needed. All 3 pages, plus a study plan built around your own exam dates.

How CapCompass uses this sheet

These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Continuity and Differentiability the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.