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Which of the following functions are continuous for all real numbers? g(x)=ln(x) f(x)=(1)/(x) Choose 1 answer: (A) g only (B) f only (C) Both g and f (D) Neither g nor f

Which of the following functions are continuous for all real numbers?\newlineg(x)=ln⁡(x) g(x)=\ln (x) \newlinef(x)=1x f(x)=\frac{1}{x} \newlineChoose 11 answer:\newline(A) g g only\newline(B) f f only\newline(C) Both g g and f f \newlineD Neither g g nor f f

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Q. Which of the following functions are continuous for all real numbers?\newlineg(x)=ln⁡(x) g(x)=\ln (x) \newlinef(x)=1x f(x)=\frac{1}{x} \newlineChoose 11 answer:\newline(A) g g only\newline(B) f f only\newline(C) Both g g and f f \newlineD Neither g g nor f f
  1. Determine Continuity: We need to determine the continuity of the functions g(x)=ln⁡(x)g(x) = \ln(x) and f(x)=1xf(x) = \frac{1}{x} for all real numbers.
  2. Consider g(x)=ln⁡(x)g(x) = \ln(x): Let's first consider g(x)=ln⁡(x)g(x) = \ln(x). The natural logarithm function ln⁡(x)\ln(x) is defined only for x > 0. Therefore, it is not defined for x≤0x \leq 0, which means it is not continuous for all real numbers.
  3. Consider f(x)=1xf(x) = \frac{1}{x}: Now let's consider f(x)=1xf(x) = \frac{1}{x}. The function 1x\frac{1}{x} is defined for all real numbers except x=0x = 0, where it has a vertical asymptote. Therefore, it is not continuous at x=0x = 0.
  4. Neither g nor f: Since neither g(x)=ln⁡(x)g(x) = \ln(x) nor f(x)=1xf(x) = \frac{1}{x} is continuous for all real numbers, the correct answer is (D) Neither gg nor ff.

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