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In the image below, the key is initially G, then changes to E (right?) for the rest of the song. However, that entire bar uses sharps from E, and the key never changes back. Why wouldn't the key change happen before that bar? Previous bars do indeed play notes from G, so it makes sense for it not to happen even sooner.
The source is "Jane's Song", by Christopher Norton.
It happens there because the previous bar isn't in E. The remaining part is, but that bar itself is the V of E, thus B major, leading into the new key. It could have been written as in the new key of E, but as the final part of the last section, it stays in G, complete with necessary accidentals.
Tradition. Time was when composers wouldn't always change key signature even when there was a clear change of key. For example, it's usual in Classical sonata-form movements for the second subject to appear first in a different key from the first subject, but the key signature to stay the same even if a change would mean easier notation. A striking example here is Beethoven's piano sonata 21 in C ("Waldstein"), where the second subject's first appearance is in E major, entailing about a hundred accidentals. In that era, composers seem to have regarded key-signature changes as suitable only at points that are important in the movement's structure. And what would indicate the close of one section and the start of another? A cadence that indicates a change of key -- as Tim suggests in his answer.
Here are two examples from the first movement of Beethoven's piano sonata 8 in c minor ("Pathétique"), where the music leading up to the key-changing cadence entails several accidentals but the key-signature doesn't change until the cadence itself.
At the start of the development, where the key is already g minor, and modulates to e minor:
In the transition from the development (which modulates through various keys) to the recapitulation (in c minor):
Composers would later get less reluctant to change key-signature (including Beethoven himself) but the legacy of this old tradition remains to some extent.
