Continuity. The Continuity Equation The equation that defines continuity at a point is called the Continuity Equation.
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1 Continuity A function is continuous at a particular x location when you can draw it through that location without picking up your pencil. To describe this mathematically, we have to use limits. Recall that a limit tells you which point the graph is heading toward. Continuity is when the graph is heading toward a point that's already filled in. The Continuity Equation The equation that defines continuity at a point is called the Continuity Equation. lim f x f a x a To understand the Continuity Equation, introduce a variable in the middle. lim f x x a b f a The left hand equality lim f x a x b says that the graph steadily approaches the point ab,. And the right hand equality f a b says that the point ab, is filled in. So, no gap at a. Continuity at a Point We say the function f is continuous at a if lim g x f x f a. x a x 1 is continuous at 1 because lim g x g 1 equal. o lim g x o f x1 says the graph is heading toward x1 g 1 says that 1, is a solid dot. x 1 x1 x1 x x1 x1 lim f x, but since 1 hole. Here we have x1 continuity equation cannot be satisfied. 1,.. They both is not continuous at 1 where is has a f is undefined, the Continuity on an Interval A function f x is continuous on the interval I if it's continuous at every point in I. The reciprocal function is continuous on the intervals,0 and 0, because it's continuous at every point in those intervals.
2 The reciprocal function is not continuous on the interval 1,1 since it's not continuous at 0. Everywhere Continuous A function f x is continuous everywhere if it's continuous on the interval, All polynomials px are continuous everywhere.. The arctan function is continuous everywhere. The "mermaid function" sin x is not continuous everywhere x because it's not continuous at 0. But it is continuous on the intervals,0 0,. and Coid Functions A function is called continuous on its domain (or coid) if it's continuous at every point in its domain. In other words, a coid function is continuous everywhere it's defined. You could say it's "continuous unto itself". (Sometimes a coid function is simply called a continuous function. We will not follow this practice in our course.) A coid function will often create discontinuities outside of its domain. For example, the reciprocal function is a coid function because it's continuous at each point in its domain. Yet outside its domain at zero, it creates an infinite discontinuity. So, even though the reciprocal function is continuous on its own domain, is not continuous at zero. It's common that a coid function will create discontinuities outside of its domain. For example, look at tan x. It's a coid function since its continuous everywhere it's defined. But it's obviously not continuous everywhere. tan x creates an infinite discontinuity at every odd multiple of where there's a vertical asymptote and tan x is not defined. Almost every function used in calculus is continuous on its domain. And all combinations of coid functions are also coid.
3 Theorem - A list of coid functions The following functions are coid. polynomials, rational functions, root functions, trigonometric functions, inverse trigonometric functions, exponential functions, and logarithmic functions. Theorem - Coid Combo Theorem A finite algebraic combination of coid functions is also continuous. Example Since x and x are both coid functions, the combo function x x is also also coid. Since it's not defined at zero where it creates a jump discontinuity, it's not continuous everywhere even though x and x are both continuous everywhere. If a function is coid, then Each piece is an unbroken curve. There are no isolated points. Wherever there is a discontinuity, the function is not defined there. Every point on its graph is embedded in a solid line. Example The Greatest Integer function is the only function on the Function Chart that's not coid. It's not coid since it's defined at the jump discontinuities it creates. If instead all the solid dots were open, then it would be a coid function. Coid functions are extremely important in mathematics because they represent the continuous processes of the real world. Finding intervals of continuity It's often important to know on what intervals a function is continuous. With coid functions, this is the same as asking what intervals are in the domain. The largest intervals on which rx and 1, because domain of rx. x 4x is continuous are, 1, 1,1, x 1 r x is a coid function and these intervals taken together are the x The function rx is continuous on [0, ) because it's a coid function and its x 1 domain includes all of [0, ). It's largest interval of continuity is 1, which is all of its domain.
4 x The continuous function r x is not continuous on [0, ] since it's not x 1 continuous at x 1 where it's not defined. Sample Test Questions on Continuity Place a T or F in the space provided. 1, x is continuous on. 1 x is continuous on 1,. x 4x 3. is continuous on 1,1. x x 11x 4x is continuous everywhere. 5. arctan x is continuous everywhere. 6. tan x creates an infinite number of discontinuities 7. The greatest integer function has an infinite number of discontinuities 8. sin xxis continuous at ln x is a coid function that creates an infinite discontinuity at zero. 10. The function x e x ln( x 1) is coid at. 11. A coid function is always continuous everywhere. 1. A coid function is always continuous everywhere it is defined. 13. A coid function is always continuous at every point in its domain. 14. If a function is continuous everywhere, then it's continuous on any interval. 15. If a function is defined at a point, then it's continuous at that point. 16. If a function is continuous at a point, then it's defined at that point. x 1, x The function f x x 1 is continuous everywhere. 1, x The natural exponential function is not continuous on its domain. 19. A rational function is continuous everywhere its denominator is not zero. 0. Polynomials are continuous everywhere. Fill in the blank with the best answer. lim f x 3 1. If f is continuous at and, then f x. If f is continuous at and f 3, then lim f x x.. x 4x 3. The largest intervals on which is continuous are. x 4 x 4. An interval containing x on which is continuous is. x 1
5 Answers T/F 1. F. T 3. T 4. T 5. T 6. T 7. T 8. F 9. T 10. T 11. F 1. T 13. T 14. T 15. F 16. T 17. F 18. F 19. T 0. T Fill in the Blank ,,,,, many correct answers including 1,3 and 1,
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