Real Analysis: Limits and Continuity of Functions

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.

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 $x = 4$.

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.

Interchange the limit of the function and solve the limit  lim_{x \to 0} \frac{\arcsin(x)}{1 + \sqrt{x}}.

Use the Intermediate Value Theorem to show that there is a root of the equation $f(x) = x^2 - x - 12$ in the interval [3, 5].

Use the Intermediate Value Theorem to find the value of $c$ in the interval $[1, 4]$ such that $f(c) = 19$, given $f(x) = 2x^2 + 3x + 5$.

Prove the Intermediate Value Theorem.

Prove that the function $f(x)$ is continuous or discontinuous at $x = 2$ and $x = 3$ using the given piecewise function: $f(x) = \sqrt{x+2}$ when $x < 2$, $f(x) = x^2 - 2$ when $2 \leq x < 3$, $f(x) = 2x + 5$ when $x \geq 3$.

Determine if the function $f(x) = 2x + 5$ when $x < -1$, $f(x) = x^2 + 2$ when $x > -1$, and $f(x) = 5$ when $x = -1$ is continuous or discontinuous at $x = -1$. If discontinuous, identify the type of discontinuity.

Prove that the square root function is continuous at $x = 0$ using the epsilon-delta definition of continuity.

Prove that the square root function is continuous at all positive numbers using the epsilon-delta definition of continuity.

If a function $f$ is continuous at a point $x_0$, then the limit of the function $f$ as $x \to x_0$ exists and is equal to $f(x_0)$. Verify this for the function $f(x) = |x|$ at $x_0 = 0$.