We study the general problem of the behaviour of the continuum function in the presence of non-supercompact strongly compact cardinals. We begin by showing that it is possible to force violations of GCH at an arbitrary strongly compact cardinal using only strong compactness as our initial assumption. This result is due to the third author. We then investigate realising Easton functions at and above the least measurable limit of supercompact cardinals starting from an initial assumption of the existence of a measurable limit of supercompact cardinals. By results due to Menas, assuming $2^{\kappa }={\kappa }^{+}$, the least measurable limit of supercompact cardinals κ is provably in ZFC a non-supercompact strongly compact cardinal which is not ${\kappa }^{+}$-supercompact. We also consider generalisations of our earlier theorems in the presence of more than one strongly compact cardinal. We conclude with some open questions.

我们研究了在非超紧强紧基数存在下连续函数行为的一般问题。我们首先表明，可以仅使用强紧凑性作为我们的初始假设，在任意强紧凑基数上强制违反 GCH。这个结果归功于第三作者。然后，我们从超紧基数的可测量极限存在的初始假设开始，研究在超紧基数的最小可测量极限及以上实现 Easton 函数。根据 Menas 的结果，假设$2^{\kappa }={\kappa }^{+}$,超紧基数κ的最小可测量极限可证明在 ZFC 中是一个非超紧强紧基基数，它不是${\kappa }^{+}$- 超紧凑型。我们还考虑在存在不止一个强紧致基数的情况下对我们早期定理的推广。我们以一些开放性问题结束。