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Let $X$ be a random variable such that:

  • $X$ is continuous with PDF $f_X$

  • $X$ has a finite moment generating function $M_x(s) = \mathbb{E}[e^{sX}] < \infty$ for all $s \in \mathbb{R_+}$

  • $X$ does not admit finite upper and lower bounds (i.e. $0 < F_X(x) < 1$ for all $x \in \mathbb{R}$

How can I show that the large-deviations function,

$$\phi(a) = \sup_{s\geq 0} \{sa - \log(M(s))\}$$

is continuous for all $a \geq 0$?

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