Q. Which of the following is a necessary condition for a function to be continuous at a point?
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A.
The function must be defined at that point
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B.
The function must be differentiable at that point
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C.
The function must be bounded
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D.
The function must be increasing
Solution
A function must be defined at a point to be continuous there.
Correct Answer: A — The function must be defined at that point
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Q. Which of the following is a necessary condition for f(x) to be continuous at x = a?
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A.
f(a) exists
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B.
lim x->a f(x) exists
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C.
lim x->a f(x) = f(a)
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D.
All of the above
Solution
All conditions must be satisfied for f(x) to be continuous at x = a.
Correct Answer: D — All of the above
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Q. Which of the following is not continuous at x = 0?
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A.
f(x) = x^3
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B.
f(x) = e^x
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C.
f(x) = 1/x
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D.
f(x) = ln(x)
Solution
f(x) = 1/x is not continuous at x = 0 as it is undefined there.
Correct Answer: C — f(x) = 1/x
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Q. Which of the following statements is true about continuous functions?
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A.
They can be differentiated everywhere
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B.
They are always bounded
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C.
They can be integrated
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D.
They can have jump discontinuities
Solution
Continuous functions can be integrated over an interval, but they are not necessarily differentiable or bounded.
Correct Answer: C — They can be integrated
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