Continuity of care, defined as an ongoing therapeutic relationship between a patient and a physician, is a defining characteristic of primary care services. However, arranging a consultation with one's regular doctor is increasingly difficult as practices seek to improve daily throughput to match rising consultation demand with an increasingly scarce supply of clinical labor. The emergence of online providers accelerates this trend. We study the productivity implications of this reduction in care continuity by analyzing consultation-level data of over 10 million office consultations in 381 English primary care practices over a period of 11 years. We find that continuity of care has a significant productivity benefit. On average, the time to a patient's next visit is 13.2% (95% CI: [12.2%, 14.1%]) longer when the patient sees the doctor they have seen most frequently over the past two years, while there is no operationally meaningful difference in consultation length. The results are consistent across several model specifications that account for confounding and selection bias. The data shows that the productivity benefit of care continuity is larger for older patients, patients with multiple chronic conditions, and patients with mental health conditions. We estimate that the total consultation demand in our sample would have fallen by 5.2% had all practices offered continuity of care at the level of the top decile of practices while prioritizing patient consultations expected to yield the largest productivity benefits. We discuss operational and strategic implications of these findings for primary care practices and for third-party payers of their services.