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Lecture 6

MATA31H3 Lecture 6: W6 Complete Lectures
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Department
Mathematics
Course
MATA31H3
Professor
Natalie Rose
Semester
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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