Suppose a customer enters into a lease- or subscription-type agreement with a business, where the business either provides a physical thing that the customer takes and uses, or the business does a thing for the customer such as mows the lawn every other week, or provides the customer with remote access to something such as internet access, web service hosting, or some online content-access service: the point is that the customer need never see or interact with any person from the business, after the agreement is reached. What passes for a written agreement simply states that the business provides that access or use of the thing to the customer for 1 year, and the customer pays a specified fee which is due up front (and was paid). There is, in particular, no clause allowing either party to terminate the contract with or without a refund (pro-rated or otherwise), no clause allows either party to unilaterally change the terms of the agreement, and no clause requires the parties to renegotiate if one side wants something.
However, there is also nothing that prohibits one party from attempting to modify the terms of the agreement. The business could propose a new term in the contract, and the customer might accept that term. I am interested in the case where the business sets forth a new term, and the customer is not interested. The customer might explicitly reject the new term; if he does not explicitly reject the term, and also does not explicitly accept it, does silence have the same effect as explicit rejection? To be concrete about the new term, assume that it is a clause about the agreement and not the service, so for example a mandatory arbitration clause, a waiver of liability, or a non-disparagement clause.
In this scenario, the customer has a continuing right to that service, and by accessing the web site, using the equipment or allowing the mowers to mow the lawn as usual, it is reasonable to assume that they are simply exercising their rights under the old contract, and have not implicitly accepted the new terms. On the other hand, it might not be inconceivable that the courts have discovered an obligation to explicitly respond, whereby failing to reject is as good as accepting. Is it a clearly settled matter of law how silence along with continued use of the paid-for service is treated? (I’m looking for a leading case citation: US law if there's a difference between US and the rest of the common law world).
There is a general Prescott v. Jones, 41 A. 352 (N.H. 1898), exemplified in insurance contracts, that
it is well settled that "a party cannot, by the wording of his offer, turn the absence of communication of acceptance into an acceptance, and compel the recipient of his offer to refuse it at the peril of being held to have accepted it"
There is a realm of "exceptions" with National Union Fire Insurance v. Ehrlich, where an insurance contract had ended and the company sent a renewal offer along with a bill (which was 2 months later rejected). The court ruled in favor of the company, citing Williston On Contracts:
Generally speaking an offeree has a right to make no reply to offers. But the relations between the parties may have been such as to have justified the offeror in expecting a reply. When property is sent to another though not ordered but under such circumstances that the latter knows that payment is expected, the silent acceptance of the property is in effect an assent to the offer of sale implied by the sending of the property.
See also Austin v. Burge, 137 S.W. 618 for renewal of a newspaper subscription. These kinds of cases fall outside of the scope of this question, for two reasons. First, they involve a customer using the service when they have no existing contractual right to do so, second, automatic renewals are commonplace and to be expected, whereas one party unilaterally changing a contract is not at all commonplace. In the situation covered by the question, the customer is still within the period of the existing contract.