f(x)=m*(E∩a,x]) 任取x1x2∈[a,b],x1x2,则 f(x2)-f(x)=m(E∩a,x2)-m(E∩an,x) km(E∩anx])+m(E⌒[x1x-m(E∩[a,x) m(E∩[x1,x2])≤m([x1,x2])=x2-x1 从而x)在[a,b]上(一致)连续; 由界值定理知,存在ξ∈[a,b],使(ξ)=c 令E1E∩a,],则E1满足要求1 2 1 2 2 1 1 1 2 1 2 1 2 1 ( [ , ]) ([ , ]) ( [ , ]) ( [ , ]) ( [ , ]) ( ) ( ) ( [ , ]) ( [ , ]) m E x x m x x x x m E a x m E x x m E a x f x f x m E a x m E a x =   = −   +  −  − =  −         从而f(x)在[a,b]上（一致）连续； 由界值定理知，存在ξ ∈[a,b] ,使f(ξ)=c, 令E1=E ∩[a, ξ]，则E1满足要求. 任取x1 ,x2 ∈ [a,b], x1<x2，则 [ a x1 x2 b ] f(x)=m*(E∩[a,x])