According to Scrooge, Titus was very stingy and traditionally tested his grandchildren with his so-called "one-pound-note test". During Scrooge's childhood, Titus would give them a one-pound-note and observed the kid's actions. If the kid kept the money, it had proven to be a real McDuck. However, if the kid decided to spend the money, like Scrooge's cousin Cyril McDuck did, it would be banished to the highlands in disgrace. Scrooge, of course, passed the test with flying colors.
In Hearts of Gold, it was revealed that Titus, a coal miner, passed the McDuck family heirloom watch onto Scrooge, for which Titus had to work in the coal mines for years. When telling this story to Huey, Dewey, and Louie, Scrooge explains that the reason why he is so attached to the watch is not because it's made of gold, but because it reminds him of his grandpa, implying that Scrooge and his grandfather had a good relationship with each other.
He was first mentioned in Carl Barks' 1955 Uncle Scrooge comic The Heirloom Watch. In this story, Scrooge tells his nephew Donald about his famous watch, that has been in the McDuck family for two hundred years. When he demonstrates the watch to Donald, he states: "It ran faithfully for my grandfather and his grandfather, and it has run faithfully for me, for many, many years!".
In the classic Barks comic The Hound of the Whiskervilles, Scrooge also mentions his grandfather: when Huey, Dewey, and Louie explain to Scrooge that, in order to obtain a high social status, "People want to know if your grandfather also wore a silk hat!", Scrooge responds: "My grandfather wore a miner's cap!", referring to his simple and impoverished youth. Don Rosa used this bit of information to create a portrait of Scrooge's grandfather for his Duck Family Tree illustration (first printed in 1993). When Don Rosa was later informed by a fan that Scrooge's grandfather was called "Titus" in earlier stories, Rosa said that, had he known about the name, he would have used it instead of "Dingus" on his Duck family tree.
In the 1960 comic The Search for Cyril (written by Bob Gregory and drawn by Tony Strobl), Scrooge's coal miner grandfather made his first actual appearance, and is referred to as Titus McDuck. Titus later appeared on a portrait on Giovan Battista Carpi's 1994 Duck family tree illustration.
In the 2017 reboot, Titus (under the name "Dirty Dingus") is among the relatives of Scrooge's whose name appears on Webby Vanderquack's board in "Woo-oo!". He later has an indirect role in "The Secret(s) of Castle McDuck!". Having found the treasure of the Knights Templar hidden inside Castle McDuck that Scrooge has been searching for, Dirty Dingus hid the treasure somewhere else and left behind a letter to his son in case he came looking for the treasure as a lesson to teach Fergus about earning money. Years after his death, Scrooge and Fergus discover his letter.