Continuity

Exercises

Consider the function $f(x) = xe^{\frac{x-1}{x}}$. a) Determine if the function is continuous in every $x \in \mathbb{R}$.
Study the continuity at $x_0 = 0$ of the function $f(x) = \begin{cases} \frac{x^2}{|x|} & x \neq 0 \\ 0 & x = 0 \end{cases}$
Find the domain of functions, compute $f(0)$, $f(-1)$, and establish whether they are even, odd or not. a) Find the domain of functions, establish if they are monotone and/or bounded. Draw their graphs.
Find the domain and classify the discontinuity points (if there are) of the following functions: a) $f(x) = \begin{cases} \frac{\sin x}{x} & x > 0 \\ e^{-x} & x \leq 0 \end{cases}$

b) $f(x) = \begin{cases} \ln(2+x) & |x| \leq 1 \\ 0 & |x| \geq 0 \end{cases}$

c) $f(x) = \begin{cases} \arctan\left(\frac{1}{x^2}\right) & x \neq 0 \\ 0 & x = 0 \end{cases}$
Classify the possible discontinuity points of the following functions: a) $f(x) = \begin{cases} \ln(2+x) & -1 \leq x < 0 \\ 0 & x \geq 0 \end{cases}$

b) $f(x) = \begin{cases} \frac{x+|x|}{x^2} & x \neq 0 \\ 0 & x = 0 \end{cases}$

c) $f(x) = \begin{cases} \arctan\left(\frac{1}{x^2}\right) & x \neq 0 \\ 0 & x = 0 \end{cases}$
Find $\alpha$ and $\beta$ such that the following functions are continuous: a) $f(x) = \begin{cases} \frac{\ln(1-2x)}{1-e^{-3x}} & \text{for } x < 0 \\ \alpha & \text{for } x = 0 \\ \beta(1+x) & \text{for } x > 0 \end{cases}$

b) $g(x) = \begin{cases} x + \frac{\ln(1+\alpha^2 x)}{x} & \text{for } x > 0 \\ \alpha + \beta x^2 & \text{for } x \leq 0 \end{cases}$
Find $\alpha$ and $\beta$ such that the following functions are continuous: a) $f(x) = \begin{cases} \frac{\ln(1-2x)}{1-e^{-3x}} & \text{for } x < 0 \\ \alpha & \text{for } x = 0 \\ \beta(1+x) & \text{for } x > 0 \end{cases}$

b) $g(x) = \begin{cases} x + \frac{\ln(1+\alpha^2 x)}{x} & \text{for } x > 0 \\ \alpha + \beta x^2 & \text{for } x \leq 0 \end{cases}$
Consider the function $f(x) = \frac{x^2}{\sin x}$. Find how to define $f(0)$ such that $f(x)$ turns out to be continuous at $x_0 = 0$.