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 Continuity Discontinuities Discontinuities: If a function is not continuous at a point in its domain, one says that it has a discontinuity there. There are different types of discontinuities: Removable discontinuity.If f(a) and are defined, but not equal. Jump discontinuity or step discontinuity, if the one-sided limit from the positive direction and the one-side limit from the negative direction are defined, but not equal. Essential discontinuity or Infinite discontinuity, One or both of the one-sided limits does not exist or is infinite. Function with a Point Discontinuity (Renovable discontinuity) $f(x)=\;\left\{\;\begin{array}\frac{x^2-9}{x-3}\;&\;\;\;,\;\;&\;x\;\neq\;3\;\\1\;&\;\;,\;&\;\;x\;=3\;\end{array}\right$ For all x-values except 3, the function is defined by the equation $y=\frac{x^2-9}{x-3}=\frac{(x+3)(x-3)}{x-3}=x+3$ Graph y=x+3. At x=3, the line has a "hole" or point discontinuity because, at this single point, $y=\frac{x^2-9}{x-3}$ has no y-value. When x=3, the functions is defined to have a value of 1. Indicate this by graphing the single point (3,1), as shown in the accompanying figure. Function With a Jump Discontinuity $f(x)=\;\left\{\;\begin{array}-x^2+5}\;\;\;\;,\;\;\;x\;<\;0\;\\\sqrt{x}\;\;\;,\;\;\;x\;\geq0\;\end{array}\right$ When x takes on negative values, f is defined by the equation f(x) = -x2+5. When x takes on non-negative values, f is defined by the equation $f(x)=\;\sqrt{x}$. At the crossover point, x=0, the function changes definition from f(x) = -x2+5 to $f(x)=\;\sqrt{x}$. Since these equations do not return the same y-value when x=0, the graph of this function has a vertical jump discontinuity, as shown in the accompanying figure: Function With Infinite Discontinuity Discuss the continuity of the function $f(x)=\;\frac{3}{x-2}$ The given function becomes positively infinite as $x\to\text{2^+}$ and negatively infinite as $x\to\text{2^-}$: $\lim_{x\to\text{2^+}}\text{f(x)}=\;+\;\infty$   and   $\lim_{x\to\text{2^-}}\text{f(x)}=\;-\;\infty$ Hence, $\lim_{x\to\text{2}}\text{f(x)}$ does not exist. The function has a vertical asymptote with an infinite discontinuity at x=2, as shown in the accompanying figure: Estudy the continuity on R of the function: The fuction is is continuous is discontinuous