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I am writing a document with some mathematics and I need help with the construction of a sentence.

Is it correct to write "the function f measures the failure of the space M to be smooth"? I am not sure if there is a mistake concerning the verb "to be", it does not sound natural to me.

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    Function f measures the extent to which the space M is not smooth would be more natural so far as actual English is concerned. I don't know exactly what your utterance is supposed to mean, but I wonder if it might be effectively equivalent to Function f measures the granularity of space M. – FumbleFingers Jul 18 at 16:20
  • Your alternative form looks good to me, I thought I may write it. However I was thinking more to how I can make my sentence correct, assuming it is not. My problem is the verb "to be", which does not sound natural to me. Granularity is not at all what I mean. – Gibbs Jul 18 at 20:13
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... failure of the space M to be smooth is indeed grammatical English, but I'm whether it's the usual way to express what you want is a question for Mathematics SE.

Usually you see an abstract quality being measured, such as "smoothness" or "roughness":

  • The function f measures the lack of smoothness of the space M

Here are a couple of examples from published papers:

Smoothness:

... certain geometric measures of smoothness ... (https://www.jstor.org/stable/3212668)

Roughness::

It is well-known that random functions are typically “rough” (non-differentiable), which raise the question of determining and measuring roughness. Probably, the most studied roughness measures are the Hausdorff dimension and the p−variation index. (https://arxiv.org/pdf/0802.0489.pdf)

Lack-of-smoothness:

Multidimensional Whittaker graduations are obtained by minimizing a linear combination of a measure of lack-of-fit between the graduated and ungraduated values, and measures of lack-of-smoothness of the graduated values across each dimension. (https://www.sciencedirect.com/science/article/pii/0167668795000216)

  • I upvoted the answer, but I think saying "the function f measures the lack of smoothness of M" does not really address my point. I used the expression "measures the failure of" because I saw this expression on a book written by an English author. I was just wondering whether the verb "to be" after it is correct in this form. – Gibbs Jul 18 at 20:09
  • I updated the answer to directly address this, hope that's better. – jonathanjo Jul 18 at 20:42
  • I just found an example on a published paper: "the tensor f measures the failure of a manifold to be (...)", so I am more positive my construction was correct. – Gibbs Jul 18 at 20:43

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