Would a triangle wave have finite or infinite sinusoidal components?

a triangle wave is continuous

Quote from here: -

The triangle wave has no discontinuous jumps, but the slope changes discontinuously twice per cycle

Having the slope change discontinuously also means an infinite range of sinusoidal components.

For instance, if you time integrated a square wave you produce a triangle wave but, all the hamonics of the original square wave are still present after the time integration: -

instructor said that since the triangle wave is continuous it can be represented by a finite number of sine

You either didn't get this right or the instructor misspoke. It's not sufficient for the signal itself to be continuous, but all derivatives must be continuous too. If there is any discontinuity in any derivative, then the repeating signal will have a infinite series of harmonics.

A triangle is continuous, but its first derivative is a square wave, which is not continuous. A triangle wave therefore has a infinite series of harmonics.

Math proof:

Take a function made up of the weighted sum of a finite series of sine/cosine components.

Its derivative is also a weighted sum of a finite series of sine/cosine components. Same if you derivate any number of times.

Since sine and cosine are continuous, the function and all its derivatives are continuous.

Thus, a function having a discontinuity in any of its derivatives can't be built with a finite series of sine/cosine components.