Let f(x) = 1/n for x = m/n with (m, n) = 1, and f(x) = 0 for irrational x. For irrational u € (0, 1) let Fu(x) = 0 if x < u and F₂(x) = 1 if x ≥ u. Show that f is Fu-integrable on [0, 1].

Q: Define f(x) = x if x is rational and f(x) = 0 if x is irrational. Compute So f dx and ſ f dx. Is f…

A:

Q: Define f : R2 → R by letting (y² – x)² (y4 + x²) f(x, y) = if (x, y) # (0, 0), and f(0,0) = 1. ||…

A: The given function is f:ℝ2→ℝ defined as: fx,y=y2-x2y4+x2if x,y≠0,01if x,y=0,0 Claim: The given…

Q: Let X ~ N (0, 1). Demonstrate that E (X)=0 and V (X)=1.

A: To demonstrate that E(X)=0 and V(X)=1 for a standard normal distribution X∼N(0,1), we need to…

Q: 2² – 2, a 3 is continuous everywhere. f(z) = O true O false

A: Given: fx=x2-2,  x≤3x+4,  x&gt;3   As we know that; The function will be continuous at x=a when,…

Q: , Use variables separation to solve the following partial deferential eq Uxx + Uyy , = 0, this…

A:

Q: Let f(x) — sin(nx) n³ n=1 Show that f is differentiable on R and that f' is continuous on R.

A: Within its interval of convergence, the derivative of a power series is the sum of derivatives of…

Q: Let f(x) be a twice differentiable function on (-o, 0) such that f"(x) is continuous on (-0, 00).…

A: Solution is in next step

Q: If f is continuous on [0, π], use the substitution u = π-x to show that π [" xf(sin(x)) dx =…

A: To prove

Q: If f is continuous on [0, π], use the substitution u = -x to show that -76 2 ["xf(s xf(sin(x)) dx =…

A:

Q: 6. Let f be a continuous function on [0, 2] whose second derivative exists on (0, 2). Assume f(0) =…

A:

Q: Let f(x)=1/x. Show that f is bounded from above on any interval (α,1), where α>0, but the function…

A:

Q: Suppose that f (0) = 2 and f' (x) ≤ 3 for x &gt; 0. Apply the MVT to the interval [0, 4] to prove…

A: