Hello, advancedfunctions2010!
Your sketch is wrong . . .
For the function: .$\displaystyle f(x) \:=\:\begin{Bmatrix}3x + 2 && x < 1 \\ x^2 && x \ge 1 \end{array}$
a) Sketch the piecewise function
b) Determine the xvalues, if any, at which the function is discontinuous.
Find appropriate limits to support your conclusion. Code:
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Obviously there is a discontinuity at $\displaystyle x = 1.$
We verify this by taking the limit of $\displaystyle f(x)$ "from both sides".
Take limit of $\displaystyle f(x)$ as $\displaystyle \,x$ approaches 1 "from below".
. . $\displaystyle \displaystyle \lim_{x\to1^}f(x) \:=\:\lim_{x\to1^}(3x+2) \;=\;1$
Take the limit of $\displaystyle f(x)$ as $\displaystyle \,x$ approaches 1 "from above".
. . $\displaystyle \displaystyle \lim_{x\to1^+} f(x) \;=\;\lim_{x\to1^+}x^2 \;=\;1$
Since the two limits are not equal, $\displaystyle f(x)$ is discontinuous at $\displaystyle x = 1.$
Edit: Too slow again . . . redundant explanation.
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