Applying the Definition of Continuity

The function $f(x)=\frac{1}{x-2}$ is not continuous at $x=2$ because it is undefined there.

A piecewise function like $\begin{cases} x-1 & x<2 \\ x+1 & x\ge2 \end{cases}$ has a jump discontinuity at $x=2$ since the one-sided limits do not match.

The function $f(x)=x^2+1$ is continuous for all $x$ since for every point, $\lim_{x \to c} f(x)=f(c)$.

A function defined by $f(x)=\begin{cases} \sin(x) & x\leq \pi \\ \cos(x) & x>\pi \end{cases}$ may fail continuity at the boundary if the values do not agree.