Let $f, g : \mathbb{R} \to \mathbb{R}$ be functions. If $g$ is continuous, then which one of the following cases implies that $f$ is continuous?
Show Hint
An odd power function like \(t^3\) is a homeomorphism of \(\mathbb{R}\)., meaning both the function and its inverse are continuous.
This ensures that taking the cube root preserves continuity.
Even powers or periodic functions lose information about the sign or interval, allowing discontinuities to be hidden.
We are given that \(g(x)\) is a continuous function. We need to find which relation between g(x) and f(x) guarantees that f(x) must also be continuous.
Step 2: Key Formula or Approach
The composition of two continuous functions is continuous.
If \(g(x) = \phi(f(x))\) is continuous, we can express \(f(x) as f(x) = \phi^{-1}(g(x)).\)
If the inverse function \(\phi^{-1}\) is continuous everywhere on \(\mathbb{R}\), then f(x) must be continuous.
If \(\phi\) is not one-to-one or its inverse is not continuous, a counterexample can exist where f(x) is discontinuous but g(x) is continuous.
Step 3: Detailed Explanation
Let us evaluate each option:
Option (A):\(g(x) = (f(x))^3\)
The function \(\phi(t) = t^3\) is strictly increasing and bijective on \(\mathbb{R}\). Its inverse \(\phi^{-1}(y) = y^{1/3}\) is continuous on all of \(\mathbb{R}\). Therefore, f(x) = (g(x))^{1/3} is continuous as a composition of continuous functions.
Option (B): \(g(x) = |f(x)|\)
Consider \(f(x) = \begin{cases} 1 & \text{if } x \ge 0 \\ -1 & \text{if } x < 0 \end{cases}\) . Then g(x) = |f(x)| = 1 is continuous, but f(x) is discontinuous at x = 0. So, this option does not guarantee continuity of f(x).
Option (C): \(g(x) = (f(x))^2\)
Using the same counterexample as in (B), g(x) = 1 is continuous but f(x) is discontinuous at x = 0. Hence, this option does not guarantee continuity.
Option (D): \(g(x) = \sin(f(x))\)
Let f(x) =\(2\pi \lfloor x \rfloor\), where \(\lfloor x \rfloor\) is the greatest integer function. Then \(g(x) = \sin(2\pi \lfloor x \rfloor)\)= 0 is continuous, but f(x) has step discontinuities at every integer. So, this option does not guarantee continuity.
Therefore, only option (A) guarantees that f(x) is continuous.
Step 4: Final Answer
The relation \(g(x) = (f(x))^3\) implies that f(x) is continuous.
