Why do cities with rich areas still contain poor areas? A cynic would attribute the disparity (and the failures to gentrify) to indifference or inattention, (or worse) disdaining neglect or ignoring, by the politicians and rich of the poor in the poor areas. 

I exemplify with cities whose disparities I have witnessed, followed by links to the maps overlaying location with income, but I exclude downtown areas which always appear rich:

New York: (Lower) Manhattan vs. The Bronx and Brooklyn (e.g. Bedford-Stuyvesant, Bushwick) 
&lt;Google search: 1, 2>

London (Boroughs): Hammersmith &amp; Fulham, Kensington &amp; Chelsea, Westminster
vs. Hackney, Tower Hamlets, and the southern boroughs like Lewisham, Croydon.
&lt;Google search: 3, 4, 5>

Toronto: Midtown (along Yonge St. subway line, like Lawrence Park) vs. Jane &amp; Finch, Rexdale
&lt;Google search: 6>

Vancouver: West &amp; North Vancouver, West Side, South Vancouver vs. Downtown East Side, Surrey
&lt;Google search: the Vancouver Sun has many maps.>
Regarding NYC, it has mostly to do with public housing and rent stabilization. 

Here's a map of public housing in NYC. Notice the large public housing developments in lower Manhattan, The Bronx, and Bushwick.

About half of all apartments in NYC are rent stabilized, and these apartments tend to be priced at a fraction of the market value and inhabited by tenants who never move (because then they'd lose their precious deals). Thus rents in NYC have a bimodal distribution.