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Let $f:\mathbb{R} \to \mathbb{R}$ be a function, what does $f(\cdot)$ mean usually? Is it another way of writing this function, or is it a real number?

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    It is to remind you that $f$ is a function that takes one (in this case, real) parameter. Unless made clear, one can mistake $f$ for a real variable. – 2017-01-25
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    This may provide an answer: http://math.stackexchange.com/questions/1286490/what-is-the-meaning-of-expressions-of-the-type-f-cdot-function-dot – 2017-01-25
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    @астонвіллаолофмэллбэрг So $f(\cdot): \mathbb{R} \to \mathbb{R}$ is this correct? I saw someone writing $f(\cdot) \in \mathbb{R}$ today and my world was turned upside down – 2017-01-25
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    @DonaldJ.Panda That's fine! The latter expression is incorrect, $f(\cdot)$ indicates a function, not a real variable. – 2017-01-25
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    @астонвіллаолофмэллбэрг This is one of those things you can die without really questioning your assumption. I remember writing $f: \mathbb{R} \mapsto \mathbb{R}$ for the longest time until someone pointed out the difference between $\mapsto$ and $\to$. – 2017-01-25
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    Aha, I do not know the difference by my admission. Do enlighten me! – 2017-01-25
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    @астонвіллаолофмэллбэрг Well a function "$f(x) = y, x,y \in \mathbb{R}$" is equivalent to $f: \mathbb{R} \to \mathbb{R}, x \mapsto y$. The $\to$ is between spaces whereas $\mapsto$ emphasize on the elements. This is the way it is written in almost all textbooks but nobody ever points out the difference! – 2017-01-25
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    Oh, I see. Thank you for pointing this out, I never knew it! – 2017-01-25
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    The difference between $\to$ and $\mapsto$ is visible in their $\TeX$-commands `\to` and `\mapsto`: The function goes from $\Bbb R$ _to_ $\Bbb R$, and $x$ is _mapped to_ $y$. I agree that the difference is subtle, but it's definitely established. – 2017-01-25
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    @Arthur I see. This wasn't my question, yet I've got something to take back. Thank you very much. – 2017-01-25

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It is another way of writing the function, emphasising that the value of $f$ at, for instance, $5$ is written as $f(5)$, and not $f5$ or $5f$ or $(5)f$ or $f|_5$ or anything else. The dot is just a placeholder.

Some would write this as $f(x)$ instead of $f(\cdot)$, but this is a slightly different emphasis again. The notation $f(x)$ tends to be associated to a specific description of $f$, for instance $f(x) = 4x-3$.

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The meaning of the symbol $f(\,)$ can be somewhat ambiguous depending upon the context.

For example, if $f(x)=x^3$ then in the equation $y=f(x)$ the meaning of $f(x)$ is $x^3$.

However, in the equation $y=f(2)$ the meaning of $f(2)$ is the number obtained by cubing 2.

So in the former equation, it represents a mathematical expression and in the latter it represents a number.

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    So the symbol $f(\cdot)$ refers to the function $f$, and the symbol $f(x)$ refers to the number that $f$ would output if we put $x$ in it? Is this the reason why people write $f(\cdot)$ instead of $f(x)$? – 2017-08-31
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    Personally, I have never found it necessary to use the notation $f(\cdot)$. If I wish to refer to the function alone, I would simply write $f$. The expression $f(x)$ can simultaneously stand for a mathematical *expression* in $x$ such as, for example the expression $x^3$ as well as the *number* resulting from performing the operations denoted by that expression. If, instead of a variable, one has a constant such as $x=2$ then $f(2)$ means the number resulting from performing the operations. – 2017-08-31
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    So if we define $f(x)=x^3$ then $f(2)$ stands for the number $2^3=8$. Now the expression $f(\cdot)$ could merely be a way of indicating that $f$ is a function of a single variable without expressing how it is defined. – 2017-08-31