They are also called linear piecewise function graphs. Step Functions: are functions that look like steps when plotted on a graph and they follow the same piecewise rules. In order for a function to be differentiable it must satisfy two conditions:Ģ) The "pieces" must match with the same slope In graphing this function we realize it is not differentiable, because when you take the derivative the two pieces do not match. The function is UNDEFINED at x = -3, you can say the graph is continuous everywhere but where it is undefined.ĭifferentiability in Regards to Piecewise Functions Can you draw this graph without lifting your pen? No, therefore the function cannot be defined as continuous. The vertical asymptote is a point on the graph where x approaches but never reaches (from left and right side). The graph below shows the continuity of the function when a = 12, notice the vertical asymptote. In order for this piecewise function to be continuous "a" must = 12. Regular functions usually apply the same process no matter which number it is given, whereas a piecewise function will look at the number first and based on that number itself and where it is found, will decide which formula to put that number into.įor example, this is a piecewise functionį ( x ) = These intervals depend on the independent variable (x) and define the function. 5 Differentiability in Regards to Piecewise FunctionsĪ piecewise function is a function which is broken up into different pieces, each piece is defined on a certain interval.4.3 Piecewise function with an asymptote.3 Continuity in Regards to Piecewise Functions.
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