Continuity

1. Determine the values of \(x\), if any, at which the function is discontinuous. At each number where \(f\) is discontinuous, state the condition(s) for continuity that are violated.  solution

2. Determine all values of \(x\) at which the function is discontinuous.  solution

(a)  \( \displaystyle{f(x)= \frac{1}{x^2-4} }\)     (b)   \( \displaystyle{f(x)= \frac{(x-2)(x+3)}{x^2-2x} }\) 

3. Find the values of \(x\) for which each function is continuous.   solution

(a)  \( \displaystyle{f(x)=x^2-3x+1} \)

(b)  \( \displaystyle{f(x)= \frac{2}{x^2+4} }\)

(c)  \( \displaystyle{f(x)=\frac{5x}{x^2-1}}\)

(d)  \( \displaystyle{f(x)= \begin{cases}  2x+3,    x \le 1\\  6x-1,    x >1  \end{cases} }\)