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What does the Continuity revision sheet cover?

Continuity at a point x=a, condition 1: f is defined at a, and on an open interval around it, condition 2: the limit as x approaches a exists, condition 3: that limit equals the function's own value at a, What an unbroken curve looks like, Continuity from the left and from the right, Worked: continuity of a three-piece function, Functions that are always continuous, constant and power functions, every polynomial, sin x and cos x, exponential and logarithm.

Is this for Class 11 Maharashtra HSC Mathematics?

Yes, it is written for the Maharashtra HSC Mathematics syllabus.

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Continuity — Class 11 Maharashtra HSC Mathematics revision notes

A 4-page visual revision sheet for Continuity, written for Class 11 Maharashtra HSC 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 revision notes, page 1 of 4 — A preview of page 1 — all 4 pages open full size in the app.

Continuity in one look

drawn without lifting the pen: three conditions, and dropping any one breaks it

Continuity at a point x=a

condition 1: f is defined at a, and on an open interval around it f(a)
condition 2: the limit as x approaches a exists lim_xto af(x)
condition 3: that limit equals the function's own value at a lim_xto af(x)=f(a)

Continuity from the left and from the right

f is continuous from the right at a if lim_xto a^+f(x)=f(a)
f is continuous from the left at a if lim_xto a^-f(x)=f(a)
continuous from both sides at a is exactly continuous at a
continuity over an interval: continuous at every interior point
plus right-continuous at a closed left endpoint
and left-continuous at a closed right endpoint

Worked: continuity of a three-piece function

Q. f(x)=3x+1 for x<2, =7 for 2≤ x<4, =x^2-8 for x≥4. Test continuity at x=2 and x=4

at x=2: f(2)=7. LHL=lim_xto2^-(3x+1)=3(2)+1=7. RHL=lim_xto2^+(7)=7
LHL=RHL=f(2)=7, so f is continuous at x=2
at x=4: f(4)=4^2-8=8. LHL=lim_xto4^-(7)=7. RHL=lim_xto4^+(x^2-8)=4^2-8=8
LHL=7 but RHL=8, so lim_xto4f(x) does not exist

Answer: continuous at x=2, discontinuous at x=4 (a jump discontinuity)

Functions that are always continuous

constant and power functions: f(x)=k and f(x)=x^n are continuous on all of R
every polynomial: built from sums and products of continuous pieces, so it stays continuous
sin x and cos x: continuous at every real x, no exceptions
exponential and logarithm: a^x continuous on R, log_bx continuous on (0,∞)

Page 1 of 4. The full sheet, all 4 pages, opens free in the app.

What this Continuity sheet covers

Continuity at a point x=a
condition 1: f is defined at a, and on an open interval around it
condition 2: the limit as x approaches a exists
condition 3: that limit equals the function's own value at a
What an unbroken curve looks like
Continuity from the left and from the right
Worked: continuity of a three-piece function
Functions that are always continuous
constant and power functions
every polynomial
sin x and cos x
exponential and logarithm
Properties of continuous functions
Three ways continuity fails at x=a
Worked: identifying a removable discontinuity
Worked: identifying a jump discontinuity
The Intermediate Value Theorem
Worked: locating a root with the Intermediate Value Theorem
Types of discontinuity at x=a, side by side
Symbols
Traps that cost marks
Tricks that save time

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