Do Fourier Transforms work with periodic monotonically increasing functions?

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When I say "work", I mean will a clear frequency be detected? And by "periodic monotonically increasing functions", I mean functions such as the stair case function.

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A frequency can be detected, but it's not so clear..

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3 staircases of 5 steps, every 15th frequency is 0. the first non-zero frequency is 3 and $15/3 = 5$. Any curious student can probably prove the general case. Some hints on that endeavor could be that

  1. the Fourier transform is linear.
  2. the Fourier transform of a box function is a sinc and
  3. the convolution theorem.