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Problem 43 Hard Difficulty

Find the numbers at which $ f $ is discontinuous. At which of these numbers is $ f $ continuous from the right, from the left, or neither? Sketch the graph of $ f $.

$ f(x) = \left\{
\begin{array}{ll}
x + 2 & \mbox{if $ x < 0 $}\\
e^x & \mbox{if $ 0 \le x \le 1 $} \\
2 - x & \mbox{if $ x > 1 $}
\end{array} \right.$

Answer

$f$ is continuous on $(-\infty, 0)$ and $(1, \infty)$ since on each of these intervals it is a polynomial; it is continuous on (0,1) since it is an exponential. Now $\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{-}}(x+2)=2$ and $\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}} e^{x}=1,$ so $f$ is discontinuous at
0. since $f(0)=1, f$ is
continuous from the right at 0. Also $\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{-}} e^{x}=e$ and $\lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1^{+}}(2-x)=1,$ so $f$ is discontinuous
at
1. since $f(1)=e, f$ is continuous from the left at 1.

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Video Transcript

Okay. Find the numbers at which F. Is discontinuous at which of these numbers is F continuous from the right. Well after a friend neither. So to do this, we're going to sketch the graph of Uh huh. Yeah. Okay. What is the graph of F going to look like? Well they're going to be a couple of divisions here. So when X is less than zero, we're going to use the top function in that piece wise which is exposed to. So less than zero. So if we have a point add zero, that's going to be an open circle because it's not less than or equal to. So it's not inclusive. Okay, so now let's plug in negative one, -1 Plus two is positive one negative two plus 20. And so on, this is just a straight linear line coming up like that. Okay, what's the next piece? E. To the X. Zero is less than or equal to X. Is less than or equal to one. Okay, what is E. To the zero? That's going to be one. What's E. To the first, you know, the first is E. Which is like 2.7 Something I believe. Okay, so we're going to have to filled in circles here because of the less than or equal to or greater than or equal to that line. Under the greater or less than that's what that means. Okay, now for greater than one, let's plug in one. Remember it's not going to be inclusive. So there's going to be an open circle here. What is 2 -1? 2 -1 is just one And then to -2 zero, two minus, three negative one. All I'm doing here is plugging in different numbers for X. And seeing what this graph looks like and this is what this graph looks like. Okay, let's find the numbers at which is discontinuous. This should be fairly obvious to spot from this graph. Okay, so our first spot is going to be at X equals zero. And the reason is that the limit from the left? Thank you. Which refers to the function X plus two. Okay. Is equal to what is it equal to 2? Whereas the limit when X approaches zero from the right, which refers to the function E to the X. Eat to the X Is equal to one. When the limit from the left doesn't match the limit from the right. We have a jump dis continuity. So at X equals zero. There is a jump to see continuity because the limit as X approaches zero from the left does not equal the limit as X approaches zero from the right, next point is going to be X equals one. And besides looking at the graph, if you just look at the piecewise, the functions, the places where the function changes, we're going to be the places that are most likely to have a discontinuity because it is a pretty big change. When a function changes like that, there's likely to be an issue with continuity. Okay, so the limit as X approaches one from the left refers to the E. To the X function. So that's just e. Which is 2.718 I believe. Don't quote me on that. Okay. And the limit when X approaches one from the right, uh we'll refer to the function two minus X. Mhm. So that is just one. So we have another gypsy jump. Sorry, jump this continuity here because when the limit from the left, okay. Mhm. Does not equal the limit from the right. That means a jump discontinuity. So quick recap. Quick recap here. The first thing we do is grab this function. So at each of these transition points we have a change but before that we have a pretty steady function. X plus two. Then you to the X. Which has dots here because it's inclusive less than or equal to zero. Less than less than or equal to one greater than or equal to zero. And then our final function of two -X. Then from there we can tell that it's a jump this continuity because it's not a smooth curve. There are differences in the points here and that's proved by the limits as expert to zero from the left and right and one from the left and right