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Finite convex functions are known to enjoy remarquable properties, but I found the word ‘’finite’’ a little bit ambiguous. I would like to know which is correct :

A convex function $f$ is finite if $f$ is finite-valued, i.e., $-\infty\lt f(x)\lt+\infty$ for all $x$.

Or

A convex function $f$ is finite if it takes finitely many values $y$ such that $f(x)=y$.

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    No real valued function (which is the context for most convex-analysis considerations) ever takes on the values $\pm\infty$, since these are not real numbers – 2017-01-14
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    @user2520938 Actually, convex functions are often allowed to take the value $+\infty$. A good example is the indicator function of a convex set. – 2017-01-14
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    @littleO I don't see how the indicator function would take on the value $\infty$, considering that its either $0$ or $1$ – 2017-01-14
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    @user2520938 such a function is in general not convex. In context of convex analysis one use an other indicator function. – 2017-01-14
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    @user251257 Ah oke. apparently I don't know much about convex analysis – 2017-01-14
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    Notice that a convex function is continuous on the interior of its proper domain. So (non constant) convex functions with an image of finite cardinality are very pathological. – 2017-01-14
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    In convex analysis, the characteristic function of a set is a convex function that is similar to the usual indicator function, but the definition bellow is better-suited to the methods of convex analysis. One can freely convert between the two.Let $\mathcal{C}$ be a subset of $\mathbb{R}^n$ that is convex. The indicator function of $\mathcal{C}$, $\delta(.|\mathcal{C})$: $\mathcal{C}$ $\to$ $\mathbb{R} \cup \{+\infty \}$ is given by : \[ $\delta(x|\mathcal{C})$ := \begin{cases} 0, & x \in \mathcal{C}; \\ + \infty, & x \not \in \mathcal{C}. \end{cases}\] – 2017-01-14

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A finite convex function is a convex function whose output is never equal to $+\infty$ or $-\infty$. For example, $f(x) = x^2$ is a finite convex function.

I think restricting the set of possible output values of the function to be a finite set would rule out most interesting examples of convex functions.

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    Thank you so much, I really appreciate your help. This definition matches the fact that if $f$ is a finite convex function throughout $\mathbb{R}^n$, then $f$ is a proper convex function on $\mathbb{R}^n$. Problem solved ! thank you – 2017-01-14