Question

If $f(g(x+y)) = f(g(x)) + f(g(y))$, $g(1)=2$, $f(2)=1$, then $g(f(x))$ is discontinuous on:

• A. $\mathbb{R}$
• B. $(0,\infty)$
• C. $(-\infty,0)$
• D. $\emptyset$

Hint:

Cauchy Functional Equation:

• $f(x+y) = f(x) + f(y)$ implies linearity under continuity.
• If $f(g(x)) = x$, then $f$ and $g$ are inverses.
• Identity function is continuous $\Rightarrow$ no discontinuities.

Answer 1

The Correct Option is D

Let $h(x) = f(g(x))$. Given: $h(x+y) = h(x) + h(y)$ \[ h(1) = f(g(1)) = f(2) = 1 \Rightarrow h(x) = x \text{ (Cauchy's eqn)} \Rightarrow f(g(x)) = x \Rightarrow g(f(x)) = x \] Then $g(f(x))$ is identity function $\Rightarrow$ continuous on $\mathbb{R}$

Answer 2

Step 1: Understand the given functional equation
We have \( f(g(x+y)) = f(g(x)) + f(g(y)) \). This resembles the Cauchy functional equation for the composition \( f \circ g \).

Step 2: Analyze the implications of the equation
Since \( f(g(x+y)) = f(g(x)) + f(g(y)) \), the function \( h(x) = f(g(x)) \) is additive.
Additive functions are known to be continuous everywhere or nowhere, depending on whether they are linear or pathological.

Step 3: Use the given values to check linearity
Given \( g(1) = 2 \) and \( f(2) = 1 \), then:
\[ h(1) = f(g(1)) = f(2) = 1 \] If \(h\) is additive and continuous at any point, it must be linear:
\[ h(x) = kx \] Since \(h(1) = 1\), we have \(k = 1\), so \(h(x) = x\).

Step 4: Conclusion about discontinuity of \(g(f(x))\)
Since \( h(x) = f(g(x)) \) is linear and continuous everywhere, it implies no discontinuity in \( g(f(x)) \).
Thus, the set of points where \( g(f(x)) \) is discontinuous is empty (\(\emptyset\)).