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in this video lets see another category and another type of always continuous functions which is known as identity function we already know and we have memorized that yes constant function is always everywhere continuous since one category we already studied the second type is identity function wherever in your question you find that you see an identity function straight away say that it is continuous you dont need to verify or prove or calculate and tell why is it so now identity function is like what it is like your Y is equal to X now you know that Y is equal to X if you want to plot the graph of y is equal to x this is x axis this is y axis now y is equal to x means what that if X is 1 Y is 1 suppose I tell you the very basics of this we have
1 2 3 4 5 here 1 2 3 4 5 here or till 4 or 5 maybe now when X is 1 Y is 1 so 1 point is 1 1 when X is 2 y is also 2 because they are equal when X is 3 y is 3 when X is 4 y is 4 and so on so basically Y is equal to X will always be a straight line if you draw it with the help of ruler of scale it will be actually very straight that means what that means you can always draw this graph without any lifting of pen or pencil did I left a pen or pencil no I just drew it like this that means no pen or pencil to be lifted that means it is always everywhere continuous similarly not only in the first quadrant but if I say my X is
minus 1 even my Y is minus 1 so it will be something like here if I go downwards to so it is minus 1 minus 1 so the first point is this similarly minus 2 minus 2 minus 3 minus 3 and even this can be extended like this so that means mod that means this graph actually directly tells me that it is for a continuous function you do not need to related with the help of limits also but still for few people who wish to verify with the help of limits I just tell you that okay let me take Y is equal to X or FX is equal to X where X can assume any real value it can be 2 it can be 3 it can be minus 3 anything right so FX is equal to X is taken and X belongs to real now for verification
purpose what I do is I take the limit X tends to a FX where I have taken a as was a as any arbitrary real as we did in the previous proof right any arbitrary real number because if we have a whole domain of real numbers we can pick up any real number right so we picked a now when you pick a you know that limit X tends to a what is the value of FX FX is nothing but X you just substitute X here now when I substituted X and I know that extends to a directly I can substitute it with the help of something called as this and my answer would be 1 it will be nothing but at a that I get because I put extends to a here similarly what is the value when f of X of f of X is X what
value of F of a F of a means say my Y is minus 1 so what is f of minus 1 since both these are equal Y is equal to X f of minus 1 will also be minus 1 similarly F of a will also be a so when F of a is also a that means the value of the function is also a the value of the limit is also a that means what that means L HL is equal to RH L is equal to value of the function at that point until always be a finite value because we are talking about real numbers so that is why I can prove from here also that my identity function is continuous everywhere and I can prove from graph also that my identity function is continuous everywhere you do not need to prove it you just need to memorize it
Identity Function, 12th class math, math
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