Real Analysis

Prove that the set of all rational numbers between 0 and 1 is countable.

Identify examples of uncountable sets within continuous subsets of the real line.

Simplify the boolean expression: $\overline{a+b}$ + $\overline{a \cdot b}$

Let $A$ be a boolean expression: $A = \overline{(A + \overline{(B \cdot C)})} \cdot \overline{(A + \overline{B})}$. Prove that $A$ simplifies to $A \cdot \overline{B}$ using DeMorgan's Theorem.

Prove that $f(x) = x$ is continuous on its entire domain, which is the real numbers.

Prove that $f(x) = \sqrt{x}$ is continuous. The domain is the non-negative reals.

Prove that $\lim_{{x \to 3}} (-2x + 1) = -5$ using an epsilon-delta proof.

Prove that the function $f(x) = |x|$ is continuous on the real numbers using the epsilon-delta definition.

Prove that if a function is differentiable at some point $C$, then it is also continuous at that point $C$.

Identify the points of discontinuity for the function $f(x) = \frac{1}{x^2}$.

Determine the value of the constant $C$ that will make the piecewise function $f(x) = \begin{cases} Cx + 3, & \text{if } x < 2 \\ 3x + C, & \text{if } x \geq 2 \end{cases}$ continuous at $x = 2$.

Find the value of the constant $a$ that will make the piecewise function $f(x) = \begin{cases} ax - 2, & \text{if } x < 3 \\ x^2 - 5, & \text{if } x \geq 3 \end{cases}$ continuous at $x = 3$.

Find the values of $A$ and $B$ that will make the function $f(x) = \begin{cases} ax + 5, & \text{if } x < 1 \\ x^2 - Bx + 9, & \text{if } 1 \leq x < 4 \\ ax^2 - Bx - 7, & \text{if } x \geq 4 \end{cases}$ continuous at both $x = 1$ and $x = 4$.

Explain why a given interval will always contain both rational and irrational numbers for any delta greater than zero.

Explain why a given interval will always contain both rational and irrational numbers for any delta greater than zero.

Identify all of the extrema on a graph given specific points or intervals, applying the extreme value theorem and recognizing whether the intervals are closed or open.

Using the Extreme Value Theorem, identify the absolute maximum and minimum values of a continuous function defined on a closed interval $[a, b]$.

Prove that $\lim_{x \to 2} (3x + 1) = 7$ using the epsilon-delta definition of a limit.

Prove that $\lim_{{x \to 5}} \sqrt{2x + 6} = 4$ using the epsilon-delta definition of a limit.

Prove that the limit of the function $f(x) = x^2 + 5x + 6$ as $x \to 1$ is 12 using an epsilon-delta argument.