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When R (i, j) = T, it is easy to verify R* i, j) = 0. So, the new function is continues at T. When R (i, j) > T, the proposed function meets the following inequality: R ( i, j ) − T ≤ | R ( i, j ) | − ( 1 + log ( 1 + | R ( i, j ) | − T ) ) − 1 ⋅ T ≤ R ( i, j ) (28).
(a) When R (i, j) = T, it is easy to verify R* i, j) = 0. So, the new function is continues at T. (b) When R (i, j) > T, the proposed function meets the following inequality: R ( i, j ) − T ≤ | R ( i, j ) | − ( 1 + log ( 1 + | R ( i, j ) | − T ) ) − 1 ⋅ T ≤ R ( i, j ) (28) .
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com