continuitycontinuity.notebook 4 october 01, 2019 determine if the following piece wise function is...
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Continuity.notebook
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October 01, 2019
Continuity
Objectives:
Determine whether a function is continuous at a number
Determine intervals on which a function is continuous
Use properties of continuity
Use IVT
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Determine whether the following are continuous or not.
a. f(x) = 3x2 5x + 4, at x = 1
b. g(x) = , at x = 2
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Determine if f(x) = is continuous at the numbers 2, 0, 2.
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October 01, 2019
Determine if the following piece wise function is continuous at 3.
x < 3
9 x = 3
x > 3
Three Types of Discontinuity
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Continuity on an Interval
A function f is continuous on an open interval (a, b) if f is continuous at every number in (a, b)
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A function f is continuous on an interval [a, b) if f is continuous on the open interval (a, b) and continuous from the right at the number a.
A function f is continuous on an interval (a, b] if f is continuous on the open interval (a, b) and continuous from the left at the number b.
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A function f is continuous on a closed interval [a, b] if f is continuous on the open interval (a, b), continuous from the right at a, and continuous from the left at b.
Continuity on a Domain
A function f is continuous on its domain if it is continuous at every number c in its domain.
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Theorems
• A polynomial function is continuous on it domain, all real numbers.
• A rational function is continuous on its domain.
• If a function is continuous on an interval, its graph has no holes or gaps on that interval.
• If a function is continuous on its domain, it will be continuous at every number in its domain; its graph may have holes or gaps at numbers that are in the domain.
• If the functions f and g are continuous at a number c, and if k is a real number, then the functions f+g, fg, f*g, and kf are also continuous at c. If g(c) ≠ =, then function f/g is continuous at c.
• If a function g is continuous at c and a function f is continuous atg g(c), then the composite function (f°g)(x) = f(g(x)) is continuous at c.
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• If f is a onetoone function that is continuous on its domain, then its inverse function f1 is also continuous on its domain.
Which one of the following is continuous?
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Determine where the following is continuous
Intermediate Value Theorem (IVT)
EXTREMELY IMPORTANT!!!!!
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Let f be a function that is continuous on a closed interval [a, b] and f(a) ≠ f(b). If N is any number between f(a) and f(b), then ther is at least one number c in the open interval (a, b) for wich f(c) = N.
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12000 ft
14000 ft
In other words, a function f that is continuous on a closed interval [a, b] must take on all values between f(a) and f(b).
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Use IVT to show that f(x) = x3 + x2 x 2 has a zero between 1 and 2.
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