Mathematics - Continuity - Differentiability Question with Solution | TestHub

Rewrite the equation as

$1+\mathrm{f}(\mathrm{x}+\mathrm{y})=(1+\mathrm{f}(\mathrm{x}))(1+\mathrm{f}(\mathrm{y}))$ Put $g(x)=1+f(x)$ to get

$$g(x+y)=g(x)g(y)$$

As $\mathrm{g}(\mathrm{x})\ge 1$, the function $\ell \mathrm{ng}(\mathrm{x})$ is defined.

Also continuous of $f$ implies continuity of $g$

Let $\mathrm{h}(\mathrm{x})=\ell \mathrm{ng}(\mathrm{x})$, we get

$$h(x+y)=h(x)+h(y)$$

The only continuous solution of this is $\mathrm{h}(\mathrm{x})=\mathrm{kx}$

$\therefore \mathrm{f}(\mathrm{x})={\mathrm{e}}^{\mathrm{kx}}+1,{\mathrm{f}}^{\prime }(0)=1$