Example of integrable, differentible sequence of functions

Is there a sequence of functions fn(x){f_n(x)} satisfying ∫10|fn|dx=1/n,and∫10|f′n|dx=1?\int_{0}^{1} |f_n| dx=1/n,\quad \mbox{and}\quad \int_{0}^{1} |f_n ‘| dx=1?
I was looking for a sequence of functions satisfying L1L^1-norm is decreasing to 00 and L1L^1-norm of derivative of functions is a positive constant.

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2 Answers
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You may try with f(x)=xn−1f(x)=x^{n-1}

  

 

I don’t think so… It does not satisfy the second condition.
– Heonjin Ha
2 days ago

  

 

OMG. It works! I was such a stupid haha.
– Heonjin Ha
2 days ago

A more “exotic” example. The graph of fn∈W1,1(0,1)f_n\in W^{1,1}(0,1) can be a strip of nn right isosceles triangles with all the hypotenuses of size 1n\frac{1}{n} along the segment [0,1][0,1]. Then
∫10|fn|dx=14n→0,and∫10|f′n|dx=1.\int_{0}^{1} |f_n| dx=\frac{1}{4n}\to 0, \quad\mbox{and}\quad \int_{0}^{1} |f_n ‘| dx=1.