Consider $f : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ defined as follows:
$$f(a,b) := \lim_{n \rightarrow \infty} \frac{1}{n} \log_{e}\left[e^{na} + e^{nb}\right]$$
For each statement, state if it is true or false.
(a) $f$ is not onto i.e. the range of $f$ is not all of $\mathbb{R}$.
(b) For every $a$ the function $x \mapsto f(a,x)$ is continuous everywhere.
(c) For every $b$ the function $x \mapsto f(x,b)$ is differentiable everywhere.
(d) We have $f(0,x) = x$ for all $x \geq 0$.