Show that if f: R→ R is continuous and f(x) = 0 for every rational number a, men f(x)= 0 for all z.

Q: Let f : R → R be a function with the following property: |f (a) – f(b)| < 4|a – b| for all a, b E R.

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Q: Let f(z) be an entire function satisfying that f(z)| ≤|z² for all z. Show that f(z) = az² for some…

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Q: If f is analytic in a neighborhood of xo and f(xo) = 0, show that f(x)/(x – xo) is analytic in the…

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Q: 5. Prove that the function f: Z → Z defined by f(x) = 3x + 1 is one-to-one.

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Q: 2. Show that if f : R → R is continuous on R and f(r) = 0 for all r E Q then f(r) = 0 for all x E R.

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Q: Use the IVT to show that f (x) = x3 + x takes on the value 9 for some x in [1, 2].

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Q: Show that the function Arg(z) is not continuous on each point of the non-positive real axis.

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Q: Let f(x) = x. Show directly that f ∊ R[0, 2]

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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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Q: Suppose f is an entire function such that there is a number M such that Re(f(z))² – Im(f(z))² <M for…

A: Given: fz is entire function. ∃M such that Re(f(z))2-Im(f(z))2≤M  .....1 As fz is entire fz2…