Problem 409 (difficulty: 4/10)

Does there exist a monotone function \(\displaystyle f\) such that

 (1)  \(\displaystyle D(f)=[0,1]\), \(\displaystyle R(f)=(0,1);\phantom{\,\cup [2,3]}\)      (2)  \(\displaystyle D(f)=[0,1]\), \(\displaystyle R(f)=[0,1]\cup [2,3]\);
 (3)  \(\displaystyle D(f)=[0,1]\), \(\displaystyle R(f)=[0,1)\cup [2,3]\);      (4)  \(\displaystyle D(f)=[0,1]\), \(\displaystyle R(f)=[0,1)\cup (2,3]\)?

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