I challenge you: it is almost never necessary to break an axis! I found this one in a recent scientific article. The axis break here is 100% unnecessary. All that was needed was a little re-scaling of the dot size and the y-axis.
(or in fact: not reporting the one number with a graph in the first place. Or: showing an almost complete pie- or bar chart).
When you cannot avoid an axis break, for example for showing highly diverged values, do it well: the break point must be clear in the axis AND in the data! Another solution is show the values along a logarithmic scale. However this works only for some data types and is almost always harder to read (make sure to then include grid lines to focus your audience right away on the unusual axis layout!).
Also, never use a design that accidentally gives the impression of a broken axis. In this example instead of a boxplot with a black bar for median, a white one was chosen, giving the impression of some interruption in the data.