Mathematics

Mathematics

Problem

Let f(x)=[x]f(x) = [x] and g(x)={0,x∈Zx2,x∈R−Zg(x) = \begin{cases} 0, & x \in Z \\ x^2, & x \in R - Z \end{cases} then (where [.][.] denotes greatest integer function) a. lim⁡x→1g(x)\lim_{x \to 1} g(x) exists, but g(x)g(x) is not continuous at x=1x = 1 b. lim⁡x→1f(x)\lim_{x \to 1} f(x) does not exist and f(x)f(x) is not continuous at x=1x = 1 c. x−1x^{-1} is continuous for all xx d. log⁡x\log x is continuous for all xx
Intuitive explanation

The key is understanding how two piecewise functions interact: ff rounds down to integers, while gg is zero only at integers and equals x2x^2 everywhere else. To find f(g(x))f(g(x)) or g(f(x))g(f(x)), you need to track what values each function produces and how the other processes them—paying careful attention to whether intermediate results land on integers or not.

Intuitive explanation

The key is understanding how two piecewise functions interact: ff rounds down to integers, while gg is zero only at integers and equals x2x^2 everywhere else. To find f(g(x))f(g(x)) or g(f(x))g(f(x)), you need to track what values each function produces and how the other processes them—paying careful attention to whether intermediate results land on integers or not.

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