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Lecture 6

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Mathematics

MATA31H3

Natalie Rose

Fall

Description

MATA 31
Week 6: Complete Lectures
Textbook: Taalman, Kohn: CALCULUS, Single Variable.
6.1 Continuity
at a Point and on an Interval.
Intuitively we thought of continuous functions
as offunctions with graphs we can sketch
"without picking up our pencil" In this section we'll develop a formal definition of
continuity using limits
DEFINITION 1. Continuity at a point.
If tion f defined
on open interval
c-p.c+p), we say that fis continuous at c if
lim f(x)- f(c)
Graph of
continuous function fis continuous at every number in an interval (has no
breaks in this interval).
The above implies:
1. fOc) is defined
(c is in the domain of fo))
3. lim f(x) f(c)
Function f
is discontinuous at c if one of the (1), (2), or 3) fails.
Types of Discontinuities
Essential Discontinuity :Infinite discontinuity
Essential discontinuity: Jump discontinuity
Removable discontinuity
DEFINITION 2. Precise definition of Continuity at a Point.
If function f defined
on open interval (c-p.c+ p, we say that fis continuous at c if
sts o such that if Ix
The restriction 00 such that
where t- g(x)
th
Combining and
Therefore
QED
fog
o, then f(g
This scheme illustrates the proof:
The number within of care taken by g to within & ofR(e) and then by fto
within Eoffug(c)
DEFINITION 4. Continuity on an Interval. Function f continuous on the open interval (a,bo if it is continuous at every number in the interval. o--------o a,b for all pe Ca,b) lim fo)- f(p Functionfcontinuous on the closed interval la bl if it is continuous on the open interval (a,b) and also continuous from the right at a and from the left at b la, bl or all p E (a,b) limfo)- fOp) f(b fa) & Limits of Constant, Identity and Linear functions. Let c, k, b ER THEOREM 1. Constant, identity and linear functions are continuous everywhere. In terms of limits limx k-k THEOREM 2 continuous: Iff and g are continuous at c, then the following functions are also 50 f/g if g (c) #0 3) cf 2) f-g THEOREM 3 functions are Polynomial, rational, root, trigonometrie, exponential and inverse continuous on their domains. THEOREM 4 If f is continuous at b and b, then lim f(g(x)-f(b) or lim f(g(x)) f(lim (x) THEOREM 5. position fog is al g(c), then th and f If g continuous at c PROOF. We need show that there exists anumber h that then fog if x We have: fis continuous at g(c) and gis continuous at c It means that the exists 5 0 such that It means that there exists oi>0 such that where t- g(x) th Combining and Therefore QED fog o, then f(g This scheme illustrates the proof: The number within of care taken by g to within & ofR(e) and then by fto within Eoffug(c)EXAMPLES
1. Find where the function f
is continuous
Domain: 20- 320
x 3 Therefore, Dom f
(3.00)
By Theorems 1-4 fi
is continuous at every point of the open interval (3,
We can check the behavior of the function from the right at po
pports the fact th
the function is not defined at point x-3)
is continuous? Evaluate
lim f(x).
2. Where fi
x3-8
-x -9 -3 x 3 herefore,
Dom f
3,2)U (2,
1. The function is continuous on both open intervals (-3
3,2)U (2,3
2. It's also continuous from the right
at x--3: lim
x3 -8
35
3. It's also continuous from the left at
x-3: lim
x -8
Therefore, f(x) is continuous on both
half-op
intervals -3,2u(2,3
To evaluate limf) we need to evaluate the left-hand and the right-hand
limits at x-2
x -8
Therefore, f(x) has infinite discontinuity at x-2.
3. Is the function f
tan(er) continuous at: i) x- ii) x ln
Dom f
der the Principal
2 2
so f(x) tan(e') is discontinuous at x-
2 2
der the Periodic fincion. Its points of discontinuity are
where 0,1,2,3
and f
tan(e') is continuous at x ez
but fi
Therefore f(x) is discontinuous at x- In
continuous?
4. On hich
interval is f
Dom f
(-1,1), so the function is continuous at (-l,1) and
discontinuous on L-1,1LI-1,1) and (-1,11 intervals.
5. Find the value of h such that f
iscontinuous at x-1
3, if
fox) is continuous at x-1 if lim f(x)- limfor)- f
lim f(r)- 3- -3-h
h +3
Therefore, f
is continuous at x-1 if h-o
EXAMPLES 1. Find where the function f is continuous Domain: 20- 320 x 3 Therefore, Dom f (3.00) By Theorems 1-4 fi is continuous at every point of the open interval (3, We can check the behavior of the function from the right at po pports the fact th the function is not defined at point x-3) is continuous? Evaluate lim f(x). 2. Where fi x3-8 -x -9 -3 x 3 herefore, Dom f 3,2)U (2, 1. The function is continuous on both open intervals (-3 3,2)U (2,3 2. It's also continuous from the right at x--3: lim x3 -8 35 3. It's also continuous from the left at x-3: lim x -8 Therefore, f(x) is continuous on both half-op intervals -3,2u(2,3 To evaluate limf) we need to evaluate the left-hand and the right-hand limits at x-2 x -8 Therefore, f(x) has infinite discontinuity at x-2. 3. Is the function f tan(er) continuous at: i) x- ii) x ln Dom f der the Principal 2 2 so f(x) tan(e') is discontinuous at x- 2 2 der the Periodic fincion. Its points of discontinuity are where 0,1,2,3 and f tan(e') is continuous at x ez but fi Therefore f(x) is discontinuous at x- In continuous? 4. On hich interval is f Dom f (-1,1), so the function is continuous at (-l,1) and discontinuous on L-1,1LI-1,1) and (-1,11 intervals. 5. Find the value of h such that f iscontinuous at x-1 3, if fox) is continuous at x-1 if lim f(x)- limfor)- f lim f(r)- 3- -3-h h +3 Therefore, f is continuous at x-1 if h-o6.3 Upper and Lower bounds. Infimum and supremum.
A set SCR is bounded

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