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If f(x) is differentiable on interval l and such that |f'(x)| ≤ a on l, then f(x) is
  • a)
     Continuous but not uniformly continuous on l
  • b)
    Uniformly continuous but not continuous on l
  • c)
    Uniformly continuous but not differentiable on l
  • d)
    Continuous, uniformly continuous and differentiable on l
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
If f(x) is differentiable on interval l and such that |f'(x)| &le...
For xy ∈ l, by Lagrange’s mean value theorem
where x < c < y
implies f(x)- f(y) = (x - y) f '(c )
implies |f(x) - f(y) | = x - y || f'(c)|
for a given  such that |f(x) - f(y)| < ε, x, y ∈ l. Hence f(x) is uniformly continuous on l. We know that every uniformly continuous function is also continuous.
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Most Upvoted Answer
If f(x) is differentiable on interval l and such that |f'(x)| &le...
For xy ∈ l, by Lagrange’s mean value theorem
where x < c < y
implies f(x)- f(y) = (x - y) f '(c )
implies |f(x) - f(y) | = x - y || f'(c)|
for a given  such that |f(x) - f(y)| < ε, x, y ∈ l. Hence f(x) is uniformly continuous on l. We know that every uniformly continuous function is also continuous.
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If f(x) is differentiable on interval l and such that |f'(x)| ≤ a on l, then f(x) isa)Continuous but not uniformly continuous on lb)Uniformly continuous but not continuous on lc)Uniformly continuous but not differentiable on ld)Continuous, uniformly continuous and differentiable on lCorrect answer is option 'A'. Can you explain this answer?
Question Description
If f(x) is differentiable on interval l and such that |f'(x)| ≤ a on l, then f(x) isa)Continuous but not uniformly continuous on lb)Uniformly continuous but not continuous on lc)Uniformly continuous but not differentiable on ld)Continuous, uniformly continuous and differentiable on lCorrect answer is option 'A'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about If f(x) is differentiable on interval l and such that |f'(x)| ≤ a on l, then f(x) isa)Continuous but not uniformly continuous on lb)Uniformly continuous but not continuous on lc)Uniformly continuous but not differentiable on ld)Continuous, uniformly continuous and differentiable on lCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If f(x) is differentiable on interval l and such that |f'(x)| ≤ a on l, then f(x) isa)Continuous but not uniformly continuous on lb)Uniformly continuous but not continuous on lc)Uniformly continuous but not differentiable on ld)Continuous, uniformly continuous and differentiable on lCorrect answer is option 'A'. Can you explain this answer?.
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