The 1847 Andover workhouse scandal triggered the creation of the Poor Law Board, a body more closely linked to the government than the Poor Law Commission, showing the government leaning towards increased centralisation as begun by the Amendment Act. This trend continued through the century with the Union Chargeability Act in the 1860s, which joined parishes together in their financial responsibilities. This change tied in with gradually changing ideas about the role of the state and could be argued to be continuous. As they were essentially changes to an existing framework, however, it is also true that they were limited.
The Liberal Reforms of 1906-11 were, however, much more comprehensive than the small legislative amendments which had predated them. While they may not have come as a shock in ideological terms (as Progressive Liberalism and the development of Socialism had been leaning towards more State provision for several decades), in terms of actual provision they represented a huge change, acknowledging for the first time that unemployment was sometimes inevitable and not a result of idleness, and providing real protection against sickness and incapacity. This causes problems for our definition of “continuity” - these significant reforms were introduced quickly, so our model of a gradually developing provision breaks down somewhat.
As to whether they were limited, of course they were – if we are discussing state provision, everything short of Communism is limited to some extent! However, if “limited” is used with the intention of meaning “limited for their purpose” or “more limited than one would expect”, we can consider its relevance to the Liberal Reforms. While they were not “unlimited”, I believe that to describe them as limited is to take away from their huge importance, as they arguably paved the way for the Welfare State. Measures introduced at that time, such as free school meals in 1906, are still in place today. It is at this point that a rival system to the Poor Law is established, further hindering our argument for continuity.
If we consider the measures taken by the National Government to combat inter-war poverty, while things for the poor had undoubtedly improved, there remained problems. With the Labour Party now an established political force, the concerns of workers and the poor generally were now much further up the political agenda, but outside circumstances had contrived to put much more of a strain on their ideas than had been anticipated. In times of emergency, political ideology must sometimes be put on the backburner. Although the all-party National Government did manage to keep some protection in place for the poor, it cut unemployment benefit to save money at a time when British industry was collapsing (although it did cut MPs' own wages, something our own government could bear in mind). It also created labour exchanges and created bodies to oversee the unemployed.
Where does this leave our definition of 'continuous but limited'? Continuity had to be interrupted, and limits had to be applied. It would seem that the accuracy of the statement varies across the period, showing that it cannot be completely accurate throughout the timespan. While there were periods of continuity, where State provision seemed to increase incrementally, events such as the Liberal reforms constituted jolts forward, perhaps dragged along by public opinion. Similarly, the emergency circumstances created between the wars left little room for change which was gradual; later, there would be another jolt forward with the creation of the Welfare State. As to how limited or otherwise this change was, I have already discussed the problematic nature of this term, but I think that when applied to how much change there was, it is inaccurate; in 1830 Britain had a system of providing for its poor based on the whims of tyrannical overseers handing out 'outdoor relief' to save the poorest from starvation, and in 1939 we had a centralised, means-tested system well on its way to becoming the State that we know today.