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copied!<p>OCaml functions can be directly mapped to and from mathematical notation. However, you need to realize that mathematical notation is ambiguous. There is not a clear distinction between the value of the function, <code>f(i)</code>, on a particular value <code>i</code>, and the function <code>f</code> itself. Often one writes <code>f(i)</code> when one means the function <code>f</code> itself. ("Let us consider a function f(i)=i+1... This function is ...") To write correct code in OCaml, you have to see clearly whether you are working with a function itself or with the value of a function.</p> <p>When you say in mathematical notation, "Consider the function <code>f(i,j)=i+g(j) where g(j)=j</code>, you are writing the values of the functions. In OCaml, this is translated to</p> <pre><code> let f i j = let g j = j in i + g j;; </code></pre> <p>or</p> <pre><code> let f = let g = fun j -> j in fun i j -> i + g j;; </code></pre> <p>If you are trying to write <code>let sum = fun i -> i + fun j -> j;;</code>, then in mathematical notation you are saying "Consider the function <code>sum</code> such that <code>sum(i) = i + g</code>, where g is a function defined by <code>g(j)=j</code>." This is mathematically incorrect: you cannot add an <em>integer value</em> <code>i</code> and a <em>function</em> <code>g</code>. You can only add an integer <code>i</code> and the <em>value</em> of the function <code>g</code> on some other integer <code>j</code>. The expression "<code>i+g</code>" is undefined, strictly speaking. Either you wanted to write <code>i + g(i)</code>, or <code>i+g(j)</code>, but not <code>i+g</code>. This is so in mathematics, and this is so in OCaml.</p>

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