91Ó°ÊÓ

A partition \( \mathcal{P} = \{ x_0, x_1, ..., x_n \} \) of an interval \([a, b]\) is called Delta-fine if for each subinterval \([x_{i-1}, x_i]\), there exists a point \( t_i \) in the subinterval such that the length of the interval from \( t_i \) to the nearest end point of the interval is less than Delta( \( t_i \) ). In other words, \( \{t_i\}_{i=1}^{n} \) is a gauging of \( \mathcal{P} \) and for each i, we have \( |t_i - x_{j(i)}| < \delta(t_i) \) where \( j(i)=i \) or \( i-1 \)

Given \(\dot{\mathcal{P}}^{\prime}\) is a Delta-fine partition of [a, c], and \(\dot{\mathcal{P}}^{\prime \prime}\) is a Delta-fine partition of [c, b], this means there exist points in each partition, \( t_i^{ \prime} \) and \( t_i^{ \prime \prime} \), such that for each i in the partitions, \( |t_i^{\prime} - x_{j(i)}^{\prime}| < \delta(t_i^{\prime}) \) in \(\dot{\mathcal{P}}^{\prime}\) and \( |t_i^{\prime \prime} - x_{j(i)}^{\prime \prime}| < \delta(t_i^{\prime \prime}) \) in \(\dot{\mathcal{P}}^{\prime \prime}\)

The union of \(\dot{\mathcal{P}}^{\prime}\) and \(\dot{\mathcal{P}}^{\prime \prime}\) forms a partition of the interval [a, b] with 'c' being one of the partition points. To show that this partition is also Delta-fine, we must verify that for each subinterval in the new partition, there exists a point for which the delta condition applies. As per the delta condition, for each subinterval, there will be a point, say \( t_i \), in the partition such that the distance from \( t_i \) to the nearest end of the subinterval is less than Delta( \( t_i \) ). This is true for all subintervals since both \(\dot{\mathcal{P}}^{\prime}\) and \(\dot{\mathcal{P}}^{\prime \prime}\) satisfy this condition. So, the union of \(\dot{\mathcal{P}}^{\prime}\) and \(\dot{\mathcal{P}}^{\prime \prime}\) is a delta-fine partition of [a, b] with 'c' as a partition point.