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For $1< s< r< t\\leq2$, we define $$H_s[0,1]=\\{f\\in C[0,1]:{\\dim}_HG_f([0,1])=s\\},$$ $$\\underline{B}_r[0,1]=\\{f\\in C[0,1]:\\underline{{\\dim}}_BG_f([0,1])=r\\}$$ and $$\\overline{B}_t[0,1]=\\{f\\in C[0,1]:\\overline{{\\dim}}_BG_f([0,1])=t\\}.$$ We prove that $H_s[0,1]\\cap\\underline{B}_r[0,1]\\cap\\overline{B}_t[0,1]$ is both strongly $\\mathfrak{c}$-algebrable and spaceable. 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