**Core Concept**
Medullary carcinoma of the breast is a rare subtype of invasive ductal carcinoma, characterized by a distinctive histopathological appearance and clinical features. This type of cancer is often associated with a better prognosis compared to other forms of breast cancer, although it can still be aggressive.

**Why the Correct Answer is Right**
Medullary carcinoma of the breast is typically a high-grade tumor with a high nuclear grade, but it often has a pushing border and a lymphocytic infiltrate, which can make it difficult to distinguish from other types of breast cancer. The presence of lymphocytes in medullary carcinoma can help differentiate it from other subtypes, as lymphocytes are often seen in this type of cancer. Additionally, medullary carcinoma often lacks estrogen and progesterone receptors, but it can be positive for HER2.

**Why Each Wrong Option is Incorrect**
**Option A:** This statement could be true, but it's not specific enough to be a defining characteristic of medullary carcinoma. Medullary carcinoma can have a variable appearance, and not all cases will have a pushing border.
**Option B:** Lymphocytic infiltrate is a key feature of medullary carcinoma, so this statement is actually true.
**Option C:** This statement is not true, as medullary carcinoma is often high-grade and can be aggressive.

**Clinical Pearl / High-Yield Fact**
Medullary carcinoma of the breast often has a distinctive histopathological appearance, with a pushing border and a lymphocytic infiltrate. This type of cancer can be challenging to distinguish from other subtypes, but the presence of lymphocytes can be a helpful clue.

**Correct Answer: C. Medullary carcinoma of the breast is often low-grade and has a favorable prognosis.**