Problem $1$

Determine if the statement is true or false. Explain your reasoning.

If a function is not continuous at a point, then it is not defined at that point.

The function

$f(x)=\{\begin{array}{cc}{x}^{2}cos(\frac{1}{x}),& x0\\ 0,& x=0\end{array}$

is continuous.

Without calculating the exact value, we know there exists a square with side length between $1$ cm and $100$ cm such that its area is $2025c{m}^2.$

If $f(x)$ is continuous and $f(x)=0$ for some xin$[a,b],$ then by the Intermediate Value Theorem, $f(a)$ and $f(b)$ have different signs.