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Problem 40 Medium Difficulty

Show that $ f $ is continuous on $ (-\infty, \infty ) $.

$ f(x) = \left\{
\begin{array}{ll}
\sin x & \mbox{if $ x < \pi/4 $}\\
\cos x & \mbox{if $ x \ge \pi/4 $}
\end{array} \right.$

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Video Transcript

This is problem number forty of the story Calculus andthe addition. Section two point five show that half its continuous on the interval from negative infinity to infinity. The function F is equal to sign of X if X is less than before and cosign of exes if X is greater than or equal to reform and we recall our definition for continuity, the limit is ex purchase A have a function f and equal to a vein. Ah, only if and only if the function f is continuous, Addie. So by showing this limit to be true everywhere on its tell me, we won't be able and its equal to the functioning value added every point on it so mean we should be able to confirm that this function is continuous on this entire domain. So we discussed that sign and co sign separately, are tricking a metric functions on their continuous on this domain already from the age of infinity to infinity, specifically sign of X. This continuous on this domain from our exit list. In my report, coastline of excess continuous on its terrain of X is greater than or equal to party before, So the main concern is whether it's continuous and it's at the point equal to power for so we seek to determine the limit. It's X approaches power for from the left and the limit has exported to power for from the right and hope that this thes two of the same and to prove our limit, which will prove, um, existence a limit between these two functions. And then we can make a statement about the function you know, Hollywood at the point. And so for the for the limiter on the left, we're using the functions time. Specifically, this function, because we're choosing a point on its domain on dysfunction, is continuous. On that, I mean, it's simply sign off perform, which is equal to skirt of two or two one in Queens with the limiter from the right Mia's cousin of X, The value we cosign a mix at apartment for on the right, which is how its domain giving a screwed up to over too. So these two limits agree with each other, and we say that this limit and six approaches power for it exists for the function, and it is equal to square two two for two And we can also show that this is the same as the function itself. After you know, you ready the value before. As we see here, F is defining point before at the coast and function go Scent of power of reform is equal to squared two for two. So we have confirmed the definition of continuity. The limit is expressed by report of F is equal to the function. You know the way data pirate port, Meaning that this function is definitely continuous at that point. And this it is continuous on the domain of all real numbers.