Functions Built from [x], x and Absolute Values

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle to start. Take two sine x, and put it inside the greatest integer brackets. Strictly between zero and two pi, two sine x hits an integer value at seven points. So the new function breaks at seven points, right? Hmm, not so fast. One of those seven points may not be a break at all. Which one, and why? Soon you will spot it in one look. Questions like this come up often in JEE, so let's begin.

0:34Where [x] and {x} break: discontinuous at integers

So, first our two tools. The square brackets mean the greatest integer less than or equal to x; from now on, just greatest integer of x. The curly braces mean the fractional part of x. x equals greatest integer of x, plus fractional part of x. The sign with a line under it means less than or equal to, and the plain one means less than: so the fractional part can be zero, but never one. Okay? Greatest integer of x stays flat, then suddenly jumps up by one. Where does it jump? At the integers, yes! Take an integer n. As x tends to n from the left, greatest integer of x tends to n minus one; the arrow means tends to. From the right, it is n itself. The two sides disagree, so the limit does not exist there. Right? And the fractional part? Rearrange the first line: fractional part of x equals x minus greatest integer of x. Continuous minus discontinuous is discontinuous, so it breaks at exactly the same points. With me? So, both greatest integer of x and fractional part of x are discontinuous exactly at the integers, and continuous everywhere else. Clear?

1:53[g(x)] with g continuous: where can it break?

Now the main idea. Take a continuous function g, with no breaks, and put it inside the brackets: greatest integer of g of x. Where can this new function break? Very good! Only where g of x is an integer. Between integer values, the greatest integer stays the same, so nothing can jump. But these points are only candidates; not every one is a break. Okay? Case one: g crosses an integer, say going upwards. Just before, its greatest integer is one less than that integer; just after, it is the integer itself. The two sides disagree: discontinuous. Got it? Now the tricky case. Suppose g just touches an integer and turns back, without crossing it. No crossing, so no break, right? No, no, careful! It depends on the side. Touching from below means g rises to the integer and falls back, like a hill top. Near that point, g is a little less than the integer, so greatest integer of g is one less, on both sides. The two sides agree with each other, but the value at the point is bigger. Discontinuous! Clear? And touching from above? Now g comes down to the integer and goes back up, like a valley floor. Nearby, g is a little more than the integer, so greatest integer of g is that same integer on both sides, and at the point. Continuous! So: crossing breaks, touching from below breaks, touching from above does not.

3:31Products: the other factor can save continuity

One more idea: products. Greatest integer of x breaks at every integer. But a product like x times greatest integer of x, or x minus one, times greatest integer of x, can be continuous at an integer. When? Think. Yes, exactly! When the other factor vanishes, that is, becomes zero at that integer. Zero times either side of the jump is zero, so the jump is flattened. Okay? Let us check one from the course: f of x equals x minus one, times greatest integer of x, at one. LHL means left hand limit: zero times zero, and the dot means times, so zero. RHL, the right hand limit: zero times one, again zero. And f of one is zero. All equal: continuous at one! Right? Now at two, x minus one is one, not zero. Left: one times one is one. Right: one times two is two. They disagree: discontinuous at two. So the factor rescues only the integer where it vanishes. Got it?

4:43Example 1: continuity of [x] + [-x]

Alright, example one. Discuss the continuity of greatest integer of x, plus greatest integer of minus x. Pause the video, and try it first. Done? Here is what many students write: the two brackets cancel, so it is zero everywhere, and continuous everywhere, right? Oh no, careful! Let us check. If x is an integer, the brackets change nothing: x plus minus x is zero. So far, so good. Okay? But try a number that is not an integer, like two point five. Greatest integer of two point five is two. And of minus two point five? Not minus two! It is minus three, the greatest integer less than or equal to minus two point five. Two plus minus three: minus one. Clear? And this happens for every x that is not an integer: the sum is minus one. Right? So stand at any integer. On both sides the value is minus one, so LHL equals RHL equals minus one, and the limit exists. But the value at the integer is zero. With me? So it is discontinuous at every integer, and since the limit exists, each one is a removable discontinuity. Two functions that jump at every integer, and their sum still breaks there, but now only the single value at the integer is out of line. That is the trap, okay?

6:13Example 2: discontinuities of [2 sin x] in (0, 2pi)

Now, back to our puzzle, at JEE level. Find the number of points in the open interval from zero to two pi where f of x, greatest integer of two sine x, is discontinuous. The round brackets mean zero and two pi are not included. Hint: list where two sine x is an integer, then decide: crossing or touching? Pause the video and try it. Done? Okay, let's check. Two sine x lies between minus two and two, so the only integer values it can take are minus two, minus one, zero, one and two. Okay? Two sine x equals one at pi over six, going up, and at five pi over six, coming down. Both are crossings, so two breaks already. Right? Two sine x equals zero at pi, and minus one at seven pi over six and eleven pi over six. All three are crossings: five breaks so far. Clear? Now, at pi over two, two sine x reaches two, its highest value, and turns back without crossing. So no break here, right? No, no, careful! It touches two from below. Just beside pi over two, two sine x is a little less than two, so f is one on both sides, and two only at the point. Discontinuous! That makes six. Got it? And at three pi over two, two sine x comes down to minus two, its lowest value, and goes back up. Which case is this, and is it a break? Touched from above, correct! Nearby, two sine x is a little more than minus two, so f is minus two on both sides, and at the point too. Continuous! This is the point that is not a break. With me? So, seven candidate points, but only six breaks: the answer is six. And our puzzle is solved: the odd one out was three pi over two, where two sine x only touched from above. See how fast that was?

8:19Quick revision: [x], {x}, crossing vs touching, products

So, quick revision. Greatest integer of x and fractional part of x break exactly at the integers. For greatest integer of g of x, with g continuous: crossing breaks, touching from below breaks, touching from above does not. And a product can stay continuous where the other factor vanishes. Now go and try the practice problems on this page. All the best!