Calculus: Continuity by: javier AWS
Calculus: Continuity
by: javier
Calculus: Continuity (continued)
by: javier
Continuity layman definition
Continuity: layman definition
Is this function continous at x = 2 ? □ Can we ’trace’ the graph around x = 2 □ without lifting our pencil □ without running out of ink
6
4
2
0 −4
−2
0
2
4
Continuity: layman definition
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
6
4
2
0 −4
−2
0
2
4
Continuity: layman definition
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
4 3 2 1 0 −4
−2
0
2
4
Continuity: layman definition
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
4 3 2 1 0 −1 −4
−2
0
2
4
Continuity: layman definition
Y
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
y = tan(x)
X
Continuity: layman definition
Is this function continous at x = π2 ? □ Can we ’trace’ the graph around x = □ without lifting our pencil □ without running out of ink
Y
y = tan(x)
X
π 2
Continuity: layman definition
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
1 0.5 0 −0.5 f(x) = sin
−1 −0.5
0
0.5
(1) x
1
Continuity rigorous definition
Continuity: rigorous definition
DEFINITION: f(x) is said to be ”continous at c” IFF? □ limx→c f(x) exists and is finite □ f(c) exists and is finite □ the above two are equal
10
5 f(x) 0
−1
0
1
2
Continuity famous types of discontinuities
Continuity: famous types of discontinuities
DEFINITION: f(x) is said to be ”continous at c” IFF? □ JUMP Discontinuities □ ASYMPTOTE Discontinuities □ REMOVABLE Discontinuities □ OTHER Discontinuities
Continuity famously continous everywhere functions
Continuity: famously continous everywhere functions
some functions that are fmaously continous at every real number: □ polynomials □ sin(x) and cos(x) □ exponential functions, ex □ combinations of these
Continuity what can you do with it
Continuity: what can you do with it
□ many nice things happen when for continous
functions □ continous functions make continous functions when added, multiplied, etc □ continous functions allow limits through
0
−2
f(x)
−4 −1
0
1
2
Continuity: what can you do with it
□ many nice things happen when for continous
functions 0
□ continous functions make continous functions
when added, multiplied, etc □ continous functions allow limits through
−2
√ √ limx→3 x + 1 = limx→3 x + 1
f(x)
−4 −1
0
1
2
Continuity: what can you do with it
functions □ continous functions make continous functions
0
when added, multiplied, etc □ continous functions allow limits through √ √ limx→3 x + 1 = limx→3 x + 1
−2
f(x)
−4 −1
0
1
2
□ many nice things happen when for continous
√ √ limx→−5 x + 1 = limx→−5 x + 1
Continuity: what can you do with it
□ EXREME VALUE THEOREM states that if a
real-valued function f is continuous in the closed and bounded interval [a,b], then f must attain a maximum and a minimum, each at least once.
0
−2
f(x)
−4 −1
0
1
2
Continuity: what can you do with it
□ EXREME VALUE THEOREM states that if a
10
real-valued function f is continuous in the closed and bounded interval [a,b], then f must attain a maximum and a minimum, each at least once.
8 6 4 2 f(x) 0 0
1
2
3
4
Continuity: what can you do with it
□ INTERMEDIATE VALUE THEOREM states that if a
5
continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.
4 3 2 1 f(x) 0 0
1
2
3
4
Continuity: what can you do with it
□ MEAN VALUE THEOREM is one of the most
important theoretical tools in Calculus. It states that if f(x) is defined and continuous on the interval [a,b] and differentiable on (a,b), then there is at least one number c in the interval (a,b) (that is a < c < b) such that the average rate of change from a to b is equal to the instantanous rate of change at point c
0
−2
f(x)
−4 −1
0
1
2
Continuity: what can you do with it
□ MEAN VALUE THEOREM (ice melting version) 0
−2
f(x)
−4 −1
0
1
2
by: javier
Calculus: Continuity (continued)
by: javier
Continuity layman definition
Continuity: layman definition
Is this function continous at x = 2 ? □ Can we ’trace’ the graph around x = 2 □ without lifting our pencil □ without running out of ink
6
4
2
0 −4
−2
0
2
4
Continuity: layman definition
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
6
4
2
0 −4
−2
0
2
4
Continuity: layman definition
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
4 3 2 1 0 −4
−2
0
2
4
Continuity: layman definition
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
4 3 2 1 0 −1 −4
−2
0
2
4
Continuity: layman definition
Y
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
y = tan(x)
X
Continuity: layman definition
Is this function continous at x = π2 ? □ Can we ’trace’ the graph around x = □ without lifting our pencil □ without running out of ink
Y
y = tan(x)
X
π 2
Continuity: layman definition
Is this function continous at x = 0 ? □ Can we ’trace’ the graph around x = 0 □ without lifting our pencil □ without running out of ink
1 0.5 0 −0.5 f(x) = sin
−1 −0.5
0
0.5
(1) x
1
Continuity rigorous definition
Continuity: rigorous definition
DEFINITION: f(x) is said to be ”continous at c” IFF? □ limx→c f(x) exists and is finite □ f(c) exists and is finite □ the above two are equal
10
5 f(x) 0
−1
0
1
2
Continuity famous types of discontinuities
Continuity: famous types of discontinuities
DEFINITION: f(x) is said to be ”continous at c” IFF? □ JUMP Discontinuities □ ASYMPTOTE Discontinuities □ REMOVABLE Discontinuities □ OTHER Discontinuities
Continuity famously continous everywhere functions
Continuity: famously continous everywhere functions
some functions that are fmaously continous at every real number: □ polynomials □ sin(x) and cos(x) □ exponential functions, ex □ combinations of these
Continuity what can you do with it
Continuity: what can you do with it
□ many nice things happen when for continous
functions □ continous functions make continous functions when added, multiplied, etc □ continous functions allow limits through
0
−2
f(x)
−4 −1
0
1
2
Continuity: what can you do with it
□ many nice things happen when for continous
functions 0
□ continous functions make continous functions
when added, multiplied, etc □ continous functions allow limits through
−2
√ √ limx→3 x + 1 = limx→3 x + 1
f(x)
−4 −1
0
1
2
Continuity: what can you do with it
functions □ continous functions make continous functions
0
when added, multiplied, etc □ continous functions allow limits through √ √ limx→3 x + 1 = limx→3 x + 1
−2
f(x)
−4 −1
0
1
2
□ many nice things happen when for continous
√ √ limx→−5 x + 1 = limx→−5 x + 1
Continuity: what can you do with it
□ EXREME VALUE THEOREM states that if a
real-valued function f is continuous in the closed and bounded interval [a,b], then f must attain a maximum and a minimum, each at least once.
0
−2
f(x)
−4 −1
0
1
2
Continuity: what can you do with it
□ EXREME VALUE THEOREM states that if a
10
real-valued function f is continuous in the closed and bounded interval [a,b], then f must attain a maximum and a minimum, each at least once.
8 6 4 2 f(x) 0 0
1
2
3
4
Continuity: what can you do with it
□ INTERMEDIATE VALUE THEOREM states that if a
5
continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.
4 3 2 1 f(x) 0 0
1
2
3
4
Continuity: what can you do with it
□ MEAN VALUE THEOREM is one of the most
important theoretical tools in Calculus. It states that if f(x) is defined and continuous on the interval [a,b] and differentiable on (a,b), then there is at least one number c in the interval (a,b) (that is a < c < b) such that the average rate of change from a to b is equal to the instantanous rate of change at point c
0
−2
f(x)
−4 −1
0
1
2
Continuity: what can you do with it
□ MEAN VALUE THEOREM (ice melting version) 0
−2
f(x)
−4 −1
0
1
2
