Real Analysis: Limits and Continuity of Functions

Give an example of a function that is discontinuous everywhere.

Assuming that $f$ and $g$ are continuous at some point $c$ in their domain $A$, prove that: 1. $K \, \cdot \, f(x)$ is continuous at $c$ for any real number $K$, 2. $f(x) + g(x)$ is continuous at $c$, 3. $f(x) \, \cdot \, g(x)$ is continuous at $c$, 4. $\displaystyle \frac{f(x)}{g(x)}$ is continuous at $c$ given that the denominator $g(x)$ is non-zero on the entire domain.

Given the function $f(x) = x$ for all $x \in [0,1)$, prove that there is no point in the interval where $f(x) = 1$, hence the supremum is not attained.

Prove that the function $\sin\left(\frac{1}{x}\right)$ is continuous but not uniformly continuous on the open interval $(0, 1)$.