Here are some notations/misconceptions students pick up in high
school, and which require modification/correction in college:

- "if a function is not defined at a point, it's discontinuous there"
  (e.g., "1/x is dis-continuous at 0")
  in particular, they'd say that -tan- is not continuous.
This is a semantic issue: a good way to address it is via "the mob"
approach: for a function to be discontinuous there must be a witness:
a specific point at which it's -defined- but not continuous.
Hence tan is continuous, b/c all the witnesses have been
"disappeared".

(the tricky semantic way of saying it is that there's a difference
between "discontinuous" and "not continuous".
is sin continuous at octopus? this doesn't make sense: it's not
defined there. Similarly, 1/x is not defined at 0, so
continuous/discontinuous don't make sense)

- working backwards
hmmm...I need a good example.
oh, here we go:
suppose (x-1)(x-2)=(x-1) -- what's x?
a naive student might divide by $(x-1)$, and get
$(x-2)=1$, so $x=3$. The problem is that the step of dividing by
$(x-1)$ is not reversible:
if $x=3$, then the statement follows
(as $(x-2)=1$, so -multiplying- you get $(x-1)(x-2)=(x-1)$)
...but if $(x-1)(x-2)=(x-1)$ it does -not- follow that $x=3$,
  because you can't just divide.
(add better examples)

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the big change is getting used to additional formality

