Continuity and IVT Practice Problems

Is $f(x)=x^2+1$ continuous at $x=2$? Verify all three conditions.

Is $g(x)=1/(x-3)$ continuous at $x=3$? Identify the first failed condition.

A function has $f(1)=4$ and $\lim_{x\to1}f(x)=4$. Is it continuous at $1$?

A function has $f(1)=4$ and $\lim_{x\to1}f(x)=5$. Which condition fails?

A function is undefined at $2$, but its limit there is $7$. Which conditions fail?

A function has unequal one-sided limits at $0$. Can it be continuous there?

Explain why existence of $f(a)$ alone says nothing about continuity.

Explain why existence of a finite limit alone does not guarantee continuity.

Find intervals of continuity of $1/(x^2-4)$.

Find intervals of continuity of $\sqrt{x-1}$.

Find intervals of continuity of $\sqrt{5-x}/(x+2)$.

Find intervals of continuity of $\ln(x+3)$.

Find intervals of continuity of $\tan x$ on $[-2\pi,2\pi]$.

Find intervals of continuity of $\sqrt{x+2}/(x^2-9)$.

Find the domain and intervals of continuity of $\ln(9-x^2)$.

Explain endpoint continuity for $f(x)=\sqrt{4-x^2}$ on $[-2,2]$.

Classify the discontinuity of $(x^2-1)/(x-1)$ at $1$.

Classify the discontinuity of $|x|/x$ at $0$.

Classify the discontinuity of $1/(x-2)^2$ at $2$.

Classify the discontinuity of $\sin(1/x)$ at $0$.

Find and classify every discontinuity of $(x^2-4)/(x^2-x-2)$.

Find and classify every discontinuity of $(x^2-9)/(x^2-6x+9)$.

Can changing one function value repair a jump? Explain.

Can changing one function value repair a vertical asymptote? Explain.

Find $c$ so $f(2)=c$ makes $(x^2-4)/(x-2)$ continuous.

Find $c$ so $f(3)=c$ makes $(x^2-9)/(x-3)$ continuous.

Find $c$ so $f(1)=c$ makes $(x^3-1)/(x-1)$ continuous.

Find $c$ so $f(4)=c$ makes $(\sqrt{x}-2)/(x-4)$ continuous.

Explain why no value at $x=0$ makes $|x|/x$ continuous there.

Explain why no value at $x=2$ makes $1/(x-2)$ continuous there.

Find $k$ so $\begin{cases}kx+1,&x<2,\\x+3,&x\ge2\end{cases}$ is continuous at $2$.

Find $a$ so $\begin{cases}2x+a,&x<1,\\x^2+3,&x\ge1\end{cases}$ is continuous at $1$.

Find $b$ so $\begin{cases}x^2+b,&x\le-1,\\3x+1,&x>-1\end{cases}$ is continuous at $-1$.

Find $m$ so $\begin{cases}mx-4,&x<3,\\2x+m,&x\ge3\end{cases}$ is continuous at $3$.

Determine whether any $k$ makes $\begin{cases}kx+5,&x<0,\\kx+2,&x\ge0\end{cases}$ continuous at $0$.

Find $a,b$ so $\begin{cases}ax+b,&x<1,\\x^2+1,&1\le x<3,\\2x+a,&x\ge3\end{cases}$ is continuous at both joins.

Design a piecewise function with one parameter that becomes continuous when the parameter is $7$.

Explain why setting left and right formulas equal is necessary but not always sufficient when the actual function value is defined by a third rule.

Show that $x^3-2x-2=0$ has a root in $(1,2)$.

Show that $x^5+x-2=0$ has a root in $(0,2)$.

Show that $\cos x=x$ has a solution in $(0,1)$.

Explain why a sign change is sufficient but not necessary for a root to exist.

Give a continuous function with a root in $(-1,1)$ whose endpoint values have the same sign.

Explain why IVT cannot be applied to $1/x$ on $[-1,1]$, despite opposite endpoint signs.

Use two bisection steps to narrow a root of $x^3+x-1$ from $(0,1)$.

Use three bisection steps to narrow a root of $x^3-2x-2$ from $(1,2)$.

Does IVT prove uniqueness? Give an example supporting your answer.

State every hypothesis and conclusion in a complete IVT solution.