State the condition for \(f\) to be continuous at \(x=a\).

Name the discontinuity when the left and right limits exist but are unequal.

Is \(f(x)=x^2+1\) continuous at \(x=3\)?

Why is \(f(x)=1/x\) not continuous at \(x=0\)?

Determine whether \(f(x)=\frac{x^2-4}{x-2}\) is continuous at \(x=2\).

For \(f(x)=\sqrt{x-1}\), state the interval on which \(f\) is continuous.

Let \(f(x)=x+1\) for \(x<0\) and \(f(x)=2x+1\) for \(x\ge0\). Is \(f\) continuous at \(0\)?

Let \(f(x)=x\) for \(x<1\) and \(f(x)=3\) for \(x\ge1\). Classify the discontinuity at \(x=1\).

Define \(f(2)\) so that \(f(x)=\frac{x^2-4}{x-2}\) for \(x\ne2\) becomes continuous at \(2\).

Find \(c\) so that \(f(x)=x^2\) for \(x<2\) and \(f(x)=cx+1\) for \(x\ge2\) is continuous at \(2\).

Use continuity to evaluate \(\lim_{x\to0}\cos(x^2+1)\).

Explain why a removable discontinuity has a limit but is not continuous.

Find all discontinuities of \(f(x)=\frac{x+1}{x^2-1}\) and classify them.

Show that \(f(x)=\frac{x^2-9}{x-3}\) can be redefined to be continuous at \(x=3\), and give the value.

Find \(a\) and \(b\) so that \(f(x)=ax+b\) for \(x<1\), \(f(x)=x^2\) for \(1\le x<2\), and \(f(x)=3x+b\) for \(x\ge2\) is continuous at \(1\) and \(2\).

For what values of \(k\) is \(f(x)=\frac{x^2-k^2}{x-k}\) removable at \(x=k\) rather than undefined everywhere near \(k\)?

Find \(c\) so that \(f(x)=\frac{x^2-cx}{x}\) for \(x\ne0\) and \(f(0)=5\) is continuous at \(0\).

A student says every function with \(f(a)\) defined is continuous at \(a\). Give a counterexample and explain.

Use the intermediate value theorem to show that \(x^3+x-1=0\) has a solution in \((0,1)\).

Explain why \(f(x)=\sin(1/x)\) is not continuous at \(0\), even if someone defines \(f(0)=0\).