City proper

A city proper is the geographical area contained within city limits. The term, "proper", is not exclusive to cities; it can describe the geographical area within the boundaries of any given locality. The United Nations defines the term as "the single political jurisdiction which contains the historical city centre."
City proper is one of the three basic concepts used to define urban areas and populations. The other two are urban agglomeration, and the metropolitan area. In some countries, city limits that act as the demarcation for the city proper are drawn very wide, in some very narrow. This can be cause for recurring controversy.
Usage.
In its strict sense, city proper is used as a technical term in demography, the statistical study of human populations. Under the title "World Urbanization Prospects", the United Nations issues every two years estimates and projections of the urban and rural populations of all countries of the world. The book defines the population of a city proper as "the population living within the administrative boundaries of a city." The book continues to say that "city proper as defined by administrative boundaries may not include suburban areas where an important proportion of the population working or studying in the city lives." 
In demography, city proper is one of the three basic concepts used to define urban areas and populations. The other two are urban agglomeration, and the metropolitan area. In addition, there are Census Statistical Areas and permutations thereof.
A United Nations University working paper titled "Urban Settlement" reviews the most commonly used data sources, and highlights the difficulties inherent in defining and measuring the size of urban versus rural populations. It says: "The city proper is determined by legal and administrative criteria, and typically comprises only those geographical areas that are part of a legally defined, and often historically-established administrative unit. However, many urban areas have grown far beyond the limits of the city proper, necessitating other measures. An urban agglomeration is the de facto population contained within the contours of a contiguous territory inhabited at urban density levels without regard to administrative boundaries’. Urban agglomerations are thus determined by density: the agglomeration ends where the density of settlement drops below some critical threshold. A still more comprehensive concept is the metropolitan area."
In short, there is no "right" manner to define a city or municipality; city proper is just one manner.
Etymology.
The term is a combination of "city" in the sense of "an incorporated administrative district", and "proper" in the sense of "strictly limited to a specified thing, place, or idea" or "strictly accurate". In encyclopedias, the term "city proper" is often used as an example to illustrate the meaning of the word "proper" in the sense of "tightly defined".
Internationalization.
Especially when translated from the English or back, "city proper" sometimes takes on different meanings in different parts of the world. Some languages have no equivalent.
The United Nations Demographic Yearbook is compiled using questionnaires dispatched annually to more than 230 national statistical offices. These questionnaires ask for the country-specific definition of urban areas, rural areas and city proper. In its glossary, the Yearbook defines "city proper" as "a locality defined according to legal/political boundaries and an administratively recognized urban status that is usually characterized by some form of local government". In its data however, the United Nations Demographic Yearbook affords the individual countries considerable leeway over the definition of "city proper". The table titled "Population of capital cities and cities of 100 000 or more inhabitants" provides several country-specific definitions for "city proper" that diverge from the provided definition:
These definitions are those given for the purpose of the United Nations Demographic Yearbook. One should not assume that these necessarily are the prevailing definitions in their respective countries.
Controversy.
In some countries, city limits that act as the demarcation for the city proper are drawn very wide, or very narrow. This can be a recurring cause for controversy.
"List of cities by surface area" lists little known cities that are larger than many countries. By area, the city of Altamira, Brazil (population 84,000) is bigger than Greece. The City of Kalgoorlie-Boulder (population 32,000) covers more area than Hungary, Portugal, Austria, or Ireland. While these examples cause limited debate, cities in China are a source of continuing contention.
The Chinese city of Hulunbuir in Inner Mongolia was recognized as the largest city in the world by area until 2018 (see image caption above). According to the "Urban Settlement" working paper, "in 1986, to cope with growing administrative demands at the local level, China essentially reclassified counties as cities in order to allow local city governments to control the surrounding areas". The authors state that "the Chinese city of Chongqing is another case in point. Even though the municipal district of Chongqing has a total population of more than 30 million inhabitants, fewer than 6 million actually live in Chongqing city proper. Depending on which classification is used, Chongqing is sometimes listed as the world's largest city and, in other cases, does not even appear in the top rung of urban population rankings". 
On the other side of the extreme is Metropolitan Lagos. This city has a population of nearly 8 million according to the latest census, which is debated. Official data by the Lagos State estimate the population of Metropolitan Lagos at more than 14 million. However, there is no Metropolitan Government. The Municipality of Lagos was disbanded in 1976 and divided into several Local Government Areas. As a result, the most populous city in Africa, and one of the most populous and fastest growing cities in the world can be missing from lists of cities proper. A similar situation exists in Australia, where large cities are divided into much smaller Local Government Areas. 