Is it illegal to go 1 mph over the speed limit?

You can be charged/ticketed with Speeding for going 1 mph over the speed limit, but so long as you sign the "promise to appear" (not an admission of guilt) on the ticket you cannot be arrested. There are two offenses in the State of Texas for which you cannot be arrested so long as you "promise to appear" and answer the charge, those being Speeding and Open Container (Alcohol).

It is illegal. However, whether one can be arrested depends on the area. In Houston, TX, they do not allow arrests for speeding unless it exceeds the speed limit by 16 mph or more OR if the person has other violations OR the person refuses to sign the tickets. That said, in some jurisdictions you can be arrested for ANY infraction. So, quit arguing with your teacher when you are wrong on the law at least generally.....

Your teacher is correct and you can be arrested if the police officer wants to arrest you.

You should listen to your teacher. It is permissible, under the federal constitution, for police to make a full custodial arrest, even for minor offenses which are punishable by a fine only. See generally, Atwater v City of Lago Vista, 532 U.S. 318 (2001). Many states have higher restrictions, and different remedies for violations of those higher restrictions, so, it may be different in Texas as a matter of state law. But, it would be permissible as a matter of federal constitutional law to be the subject of a full, custodial arrest for a speeding violation.