Theoretical Recall
A function $f$ is continuous at $x_0$ if
\[\lim_{x \to x_0} f(x) = f(x_0).\]Equivalent $\varepsilon$–$\delta$ definition:
for all $\varepsilon>0$ there exists $\delta>0$ such that
$|x-x_0|<\delta \implies |f(x)-f(x_0)|<\varepsilon$.
Types of discontinuities:
- Removable: limit exists, but $f(x_0)$ missing or mismatched.
- Jump: one-sided limits finite but different.
- Infinite: one or both one-sided limits $\pm\infty$.
- Oscillatory: limit does not exist due to oscillation.
Basic facts:
- Polynomials, rational functions (on their domain), $\sin,\cos,e^x,\log x$ are continuous on their domains.
- Algebra and composition of continuous functions yield continuous functions.
Exercises
Exercise 1
Check continuity of \(f(x)=\frac{x^2-1}{x-1}, \quad x\ne1, \quad f(1)=c.\)
Solution:
For $x\ne1$, $f(x)=x+1$. Limit at $1$ is $2$. To be continuous, set $f(1)=c=2$.
Final Result: \(c=2 \quad \text{makes $f$ continuous at } x=1.\)
Exercise 2
Classify discontinuity of $\operatorname{sgn}(x)$ at $0$.
Solution:
$\lim_{x\to0^-}=-1$, $\lim_{x\to0^+}=1$. Different finite limits.
Final Result: \(\text{Jump discontinuity at } x=0.\)
Exercise 3
Study $f(x)=\frac{1}{x}$ at $x=0$.
Solution:
One-sided limits diverge to $\pm\infty$.
Final Result: \(\text{Infinite discontinuity at } x=0.\)
Exercise 4
Study $f(x)=\frac{\sin x}{x}$ for $x\ne0$, with $f(0)=1$.
Solution:
Limit $\lim_{x\to0}\frac{\sin x}{x}=1=f(0)$. Continuous at $0$.
Final Result: \(f \text{ continuous on } \mathbb{R}.\)
Exercise 5
$f(x)=|x|\sin(1/x)$ for $x\ne0$, $f(0)=0$. Study continuity at $0$.
Solution:
$|f(x)|\le |x|\to0$, so $\lim_{x\to0}f(x)=0=f(0)$.
Final Result: \(f \text{ is continuous at } 0.\)
Exercise 6
$f(x)=
\begin{cases}
ax+b,&x<2,
x^2,&x\ge2
\end{cases}$.
Choose $a,b$ for continuity at $2$.
Solution:
$\lim_{x\to2^-}=2a+b$, $f(2)=4$. Condition: $2a+b=4$.
Final Result: \(\text{Continuity at }2 \iff 2a+b=4.\)
Exercise 7
Check continuity of $f(x)=\log(x^2-4)$.
Solution:
Domain requires $x^2-4>0 \iff |x|>2$. On domain, composition of continuous functions is continuous.
Final Result: \(f \text{ continuous on } (-\infty,-2)\cup(2,\infty).\)
Exercise 8
$f(x)=
\begin{cases}
\sin(1/x),&x\ne0,
0,&x=0.
\end{cases}$
Check continuity at $0$.
Solution:
Limit $\lim_{x\to0}\sin(1/x)$ does not exist (oscillations). So discontinuous at $0$.
Final Result: \(\text{Oscillatory discontinuity at } 0.\)
Exercise 9
$f(x)=
\begin{cases}
x^2,&x\le0,
1,&x>0.
\end{cases}$
Check continuity at $0$.
Solution:
Left limit $0$, right limit $1$. Different.
Final Result: \(\text{Jump discontinuity at } 0.\)
Exercise 10
$f(x)=
\begin{cases}
\cos x,&x<0,
a+bx,&x\ge0.
\end{cases}$
Find $a,b$ for continuity at $0$.
Solution:
$\lim_{x\to0^-}\cos x=1$, $f(0)=a$. Continuity $\Rightarrow a=1$. $b$ free.
Final Result: \(a=1,\; b \in\mathbb{R}.\)