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Continuity of Function at a Number
A function y = f(x) is continuous at a number a if and only if the following are satisfied: If one or more of these three conditions are not satisfied, then the function f(x) is said to be discontinuous at a.
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equal f is continuous at x = 1.
Graphically, it means there is no break in the graph of f(x) at x = 1.
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h is not continuous at x = 1.
Not equal h is not continuous at x = 1. Graphically, it means there is a break ( jump) in the graph of h(x) at x = 1.
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f is not continuous at x = 2.
Graphically, it means there is a break (hole) in the graph of f(x) at x = 2.
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h is not continuous at x = -2.
Graphically, it means there is a break in the graph of h(x) at x = -2 (vertical asymptote).
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x = -2 (vertical asymptote) y = 2 is a horizontal asymptote
- 4 - 3 - 1 y = h(x) 7 12 - 8 (0, -3) , (3, 0) vertical asymptote x = -2 y = 2 is a horizontal asymptote 9
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Types of Discontinuity
Removable discontinuity – discontinuity where Graphically, it means there is a break (hole) in the graph of f(x) at x = a and this hole can be patched by redefining f(x) to be equal to its limit L when x = a. Redefine
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f has removable discontinuity at x = 2.
Redefine
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Types of Discontinuity
b) Essential discontinuity – discontinuity where (i) Jump discontinuity – essential discontinuity where Graphically, it means there is a break (jump) in the graph of f(x) at x = a. The graph approaches two different points as x approaches a from the left and right.
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Not equal h has a jump essential discontinuity at x = 1.
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Types of Discontinuity
b) Essential discontinuity – discontinuity where (ii) Infinite discontinuity – essential discontinuity where Graphically, it means there is a break in the graph of f(x) at x = a where x = a is a vertical asymptote.
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h has an infinite essential discontinuity at x = -2.
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x = -2 (vertical asymptote) y = 2 is a horizontal asymptote
- 4 - 3 - 1 y = h(x) 7 12 - 8 (0, -3) , (3, 0) vertical asymptote x = -2 y = 2 is a horizontal asymptote 19
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