How to write Vimscript functions that operate on a visually selected area

I was trying to write a Vim function that would act on a visual selection but found it surprisingly difficult.

As an example, suppose we want to write a function that converts a visual selection to upper case and map it to the ,u key binding¹. A first attempt might be:

function! MakeUpperCase() range
    U  " or `normal! U`
endfunction

" Define visual-mode mapping
xnoremap ,u call MakeUpperCase()<cr>

but hitting ,u in visual mode errors:

E464: Ambiguous use of user-defined command U

This is because the function body is executed in normal mode (which can be verified by checking the value of mode() in the function body).

So we need to reselect the visual area first using the < and > marks that are set when the function is called:

function! MakeUpperCase() range
    normal! `<v`>U
endfunction

This works.

But suppose we want a function that upper cases the contents of the paragraph under the cursor. You might think this would work:

function! MakeCurrentParagraphUpperCase()
    " Select current paragraph
    normal! vip
    call MakeUpperCase()
endfunction

" Define a normal-mode mapping
noremap ,p call MakeCurrentParagraphUpperCase()<cr>

But this fails as MakeUpperCase is being called in visual mode.

So we extend MakeUpperCase to check the value of mode() to determine the appropriate behaviour:

function! MakeUpperCase()
    if mode() == "v"
        normal! U
    else
        normal! `<v`>U
    endif
endfunction

This works in both cases.