Modified propylene polymer, composition containing the same and use thereof

A propylene polymer, comprising:

• a propylene polymer main chain having a stereo-block structure containing an isotactic block, and
• a side chain containing a carboxylic acid group, an acid anhydride group or a carboxylic acid ester group,
• wherein when said propylene polymer is dissolved in toluene to a concentration of 10% by weight at 25° C., the insoluble matter is 1% by weight or less based on the total weight of the polymer;
• wherein
• when peaks originated from the carbon atom of methyl group in a propylene unit chain region comprising a head to tail bond are measured for the propylene polymer main chain by 13C-NMR, and
• when chemical shift of the peak top of a peak assigned for a pentad represented by mmmmm is defined as 21.8 ppm,
• a ratio of area S1 of a peak having a peak top of 21.8 ppm to the total area S of peaks found at from 19.8 ppm to 22.2 ppm (S1/S) is 10% or more and 60% or less.