Determine whether the following functions are continuous at given number \(a\). If there is a discontinuity determine if it is removable or non-removable.

1. \(\displaystyle  f(x) = \left\{ \begin{array}{lr}  x^2+1 & ,  x \leq 0\\  x^3+1 & ,  x > 0  \end{array} \right. \)         at         \( a=0\)     Solution



2.\(\displaystyle  f(x) = \left\{ \begin{array}{lr}  \frac{x^2-9}{x-3} & ,  x \ne 3\\  x+2 & ,  x = 3  \end{array} \right. \)         at        \( a=3\)  Solution



3. \(\displaystyle  f(x) = \left\{ \begin{array}{lr}  3\cos{(x-5)} & ,  x < 5\\  2 & ,  x = 5  \\ e^{x-5}+2 & ,  x >5 \end{array} \right. \)        at      \( a=5\)  Solution



4. From the graph of \(f\), state the numbers at which \(f\) is discontinuous and explain why. For each of the discontinuities determine whether \(f\) is continuous from the right, from the left or neither. Also determine whether each discontinuity is removable or non-removable.   Solution


 


5. For what value of the constant \(c\) is the function \(f\) continuous on \((-\infty, \infty)\)?      

\(\displaystyle  f(x) = \left\{ \begin{array}{lr}  cx^2+3 & , x \leq 2\\  x^3+c & , x > 2  \end{array} \right. \)      Solution