Let f be a real-valued function whose domain is a subset of R. Then f is continuous at x_0 in dom(f) if and only if for each e > 0 there exists δ > 0 such that x ∈ dom(f) and |x − x_0| < δ imply |f(x) − f(x0)| < e. Use the defintion above to prove that: f(x) = x(x-1) if x is rational                                                                       = 0 if x is irrational  is continuous at x = 0 and x =1 and discontinuous everywhere els

Q: Verify that the three conditions of the hypothesis of Rolle's Theorem hold for the function f (x) =…

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Q: Sketch the graph of a function f such that f (x) ≥ 0 on [0, 1] and f (x) ≤ 0 on [1, 2], whose…

A: Consider the function Then and The graph of the function is shown below,

Q: let f be the function x²-C for x< 5 flx)=3x+5c for X Z 5

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Q: (3-x, xs1 Consider the function given by f (x) = Is the function continuous, %3D x -x+1, x>1…

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Q: If f is a function that is continuous on the open interval (1, 4), then f is continuous at x = 3.…

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Q: Find and classify the extreme values of f(r)=6r 4r over the interval (-2 21

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Q: 6. Verify Mean Value Theorem for the function f(x) = 3x² -4x +5 in [-1,1].

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Q: 6. Prove that f(x)=√x is continuous for all x ≥ 0.

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Q: Let f: X --> Y be a function, and let U be a subset of X.  (1) Prove or disprove that f maps X \ U…

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Q: Which of the following interval is f(x) = O (2, +∞0) O [2, +∞0) 0 (-∞0, +∞0) ○ (0, +∞) 1 Vx…

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Q: If f is continuous on the interval [5, 12] and f(5) = 7 and f(12) = -2, then f must have at least…

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.