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 else 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(x_0)| < e.

Q: Consider the function f : defined by                    f(x) = 0 if x is irrational and f(x)=1 if x…

A: The objective of the question is to determine whether the given function f, which is defined…

Q: Use the intermediate value theorem to classify each function according to the existence of zeros on…

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Q: Let f (x) = x2 if x is rational and f (x) = 0 if x is irrational. b) Prove that f is differentiable…

A: Given fx=x2x is rational0x is irrational To prove f is differentiable at exactly one point, namely…

Q: Let f: R → R such that f(x) = -x if x is rational and f(x) = 2x+3 if x is irrational. Prove that f…

A: fx=-xif x is rational2x+3if x is irrational We have to show f is continuous at a iff a=-1

Q: Suppose that f is a function which is integrable on the entire real line, and that a and b are real…

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Q: Let f RR such that f(x) = -x if x is rational and f(x) = 2x+3 if x is irrational. Prove that f is…

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Q: Define f: R -R by Sir +1 if r 4. Find all real values at which f(x) is continuous. Prove that your…

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Q: Consider the function f with domain (-2, : 3|22 – 1| if – 2 < x < 1 2x2 + 2x – 4 f(x) = 3 if x = 1…

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Q: Suppose that the function f: [0, 1] in R is continuous and thatf(x) &gt; 2      if 0&lt;=X &lt; 1.Is…

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Q: Show that the function r(0) = sec 0- + 5 has exactly one zero in the interval 0,5 every point over…

A: Given:-               Show that the function r(θ) = sec θ-1θ3+5- has exactly one zero in the…

Q: 2. Let f(x) = 1 for rational x and f(x) = 0 for irrational x. Show that if F is increasing and…

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Q: 8. Suppose that the function f: [0,1] → R is continuous and its image consist en- tirely of rational…

A: Kindly refer to Step 2 for the solution.

Q: Which of the following functions f: R → R is continuous at the point 0 € R? f(x) = x² f(x) = = X {²…

A: The given functions,(i) (ii) (iii) (iv) (v) The aim is to check the continuous of the function at…

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.