The Bug:
Block breaking progress can be retained by using the world border.
Steps to Reproduce:
Locate the world border and find a block that's on the edge of it.
Begin breaking the block and without letting go of your Attack/Destroy button, move your focus to the world border.
Take note as to whether or not block breaking progress can be retained by using the world border.
Observed Behavior:
Block breaking progress can be retained.
Expected Behavior:
Block breaking progress would not be retained.
Linked issues
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Confirmed for 14w19a.

Confirmed for 14w21b
Confirmed for 1.8pre2

Confirmed for 1.9

Still in 16w15b. Also you can save it with unholded LMB. You should while breaking the block look at the border and unhold the button at the same time. I can't reproduce it consistently, but sometimes it works.
Confirmed for 1.13.1.
If the current reporter (I don't know if he is active) doesn't mind, I'd like to be the reporter of this issue, I'll update it accordingly.
Done
Confirmed for 1.16 Pre-release 2.
Can confirm in 20w48a.
Can confirm in 21w03a.
Can confirm in 21w05b.
Can confirm in 21w06a.
Can confirm in 21w07a.
Video attached.
Can confirm in 21w17a.
Can confirm in 1.17.
Confirmed.
Breaking animation vanishes if looking at Worldborder + 2 blocks.