Does the KJV fail to translate the Greek article properly?

It is often alleged that the KJV erroneously translates the Greek definite article (ο, η, τό) as an English indefinite article (a, an). An example is in Matthew 5:1: “he went up into a mountain.” The Greek says, “ανεβη εις το ορος,” which has the definite article “το” preceding “mountain (ορος).” The KJV is not in error. The definite article in Greek can function as a categorical article having a qualitative force (Daniel Wallace, Greek Grammar Beyond the Basics at 228), in which case the English indefinite article could be a valid translation. As with Greek, the English definite article (the) can also be used to determine the category and nothing more. For example, we might say “On sunny days, people go to the beach.” Despite the definite article, no specific beach is implied. Thus we are actually saying, “On sunny days, people go to a beach.” “The beach” is a categorical determination, not a determination of a specific beach. The definite article’s purpose is only to determine the specific category, and not to determine the specific thing in the category. Likewise, when Matthew 5:1 says, “ανεβη εις το ορος,” “το” can be translated with the English indefinite article, signifying that the category of the location was a mountain as opposed to something else (e.g. town, beach). The NIV, which attempts to convey the sense of the passage, agrees with the KJV and reads, “he went up on a mountainside.”