TITLE: How to show that $f(x)=x^2$ is continuous at $x=1$?
QUESTION [25 upvotes]: How to show that $f(x)=x^2$ is continuous at $x=1$?

REPLY [7 votes]: The product of continuous functions is continuous. The function $x$ is continuous, hence also $x^2=x\cdot x$ is continuous.
The proof that the product of continuos functions is continuous, simply relies on the theorem that states the limit of the product is the product of the limits.