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 Pre-algebra Arithmetics Integers Divisibility Decimals Fractions Exponents Percentages Proportional reasoning Radical expressions Graphs Algebra Monomials Polynomials Factoring Linear Equations Graphs of linear equations Rectangular Coordinate System Midpoint Formula Definition of Slope Positive and negative slope Determine the slope of a line Equations of lines Equation of lines (from graph) Applications of linear equations Inequalities Quadratic equations Graphs of quadratic equations Absolute Value Radical expressions Exponential equations Logarithmic equations System of equations Graphs and functions Plotting points and naming quadrants Interpreting Graphs Relations and Functions Function Notation Writing a Linear Equation from a Table Writing a Linear Equation to describe a Graph Direct Variation Indirect Variation Domain and range Sequences and series Matrices Inverse of a matrix Determinants Inner product Geometry Triangles Polygons 2-D Shapes 3-D Shapes Areas Volume Pythagorean Theorem Angles Building Blocks Geometry Transformations Parallel, coincident and intersepting lines Distances in the plane Lines in space Plane in space Angles in the space Distances in the space Similarity Precalculus Sequences and series Graphs Graphs Definition of slope Positive or negative slope Determine the slope of a line Equation of a line (slope-intercept form) Equation of a line (point slope form) Equation of a line from graph Domain and range Quadratic function Limits (approaches a constant) Limits (approaches infinity) Asymptotes Continuity and discontinuities Parallel, coincident and intersepting lines Introduction to Functions Limits Continuity Asymptotes Trigonometry Trigonometric ratios The reciprocal trigonometric ratios Trigonometric ratios of related angles Trigonometric identities Solving right angles Law of sines Law of cosines Domain of trigonometric functions Statistics Mean Median Mode Quartiles Deciles Percentiles Mean deviation Variance Standard Deviation Coefficient of variation Skewness kurtosis Frequency distribution Graphing statistics & Data Factorial Variations without repetition Variations with repetition Permutations without repetition Permutation with repetition Circular permutation Binomial coefficient Combinations without repetition Combinations with repetition

 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