Video 19
Continuity of Elementary Functions
Basic Elementary Functions: Finite Operations, which can be expressed in one formula
Basic Elementary Functions are Continuous in their Definitions
1. Continuity of Sums, Products, and Quotients of Continuous Functions
(1) Finite The algebraic sum of a function that is continuous at a certain point is still a function that is continuous at that point
(2) a finite number of functions that are continuous at a certain point and their product is still a function that is continuous
at a certain point (3) two are continuous at a certain point Their quotients are still functions that are continuous at this point, as long as the function value of the denominator at this point is not equal to 0.
Prove (3)
Let f(x), g(x) be continuous at the point x0,
then there is lim(x ->x0) f(x) = f(x0),lim(x->x0)=g(x0) <>0
Prove the continuity of trigonometric functions
Example y=sinx , y=cosx at (-∞, +∞ ) is continuous in
y=tanx, y=cotx is continuous in its domain.
Prove that tangent and cotangent function
y = tanx = sinx/cosx
y = cotx = cosx/sinx
Using the continuity of the continuous function quotient, it can be seen that tanx and cotx are in continuous within the domain of definition.
Exercises 1, 2, 3, 5, 6, 7
2 Continuity of inverse and composite functions
If function y = f(x) is continuous on the interval Ix, (monotonically increasing or monotonically decreasing), then its inverse x=φ(y) must exist
, and it is also monotonically increasing (decreasing) and continuous on the corresponding interval Iy = {y|y=f(x), x belongs to Ix}
For example: y=sinx is monotonically increasing and continuous on Ix [-π/2,π/2] , so its inverse function y=arcsinx is monotonically increasing on [-1,1] and is continuous.
y=cosx is monotonically decreasing on Ix[0,π], so its inverse function is monotonically decreasing in the interval [-1,+1] and continuous
y=tanx is monotonically decreasing in (-π/2,π/2) increases and is continuous, its inverse function is monotonically increasing in y=arctanx in the (-∞, +∞) interval and continuous
y=cotx is monotonically decreasing and continuous in (0, π), then its inverse function y=arccotx is in ( -∞, +∞) monotonically decreasing and continuous
(2) Let when x->x0, u=φ(x) limit exists, and lim(x->x0) φ(x) = a, and y = f (u) is continuous at the corresponding point u= a,
then when x->x0, the limit of the review function f[φ(x)] exists, and lim(x->x0)f[φ(x)] =
Proof of f(a) :
For example: Find lim(x->0) ln(1+x)/x
Continuity of Elementary Functions
Basic Elementary Functions: Finite Operations, which can be expressed in one formula
Basic Elementary Functions are Continuous in their Definitions
1. Continuity of Sums, Products, and Quotients of Continuous Functions
(1) Finite The algebraic sum of a function that is continuous at a certain point is still a function that is continuous at that point
(2) a finite number of functions that are continuous at a certain point and their product is still a function that is continuous
at a certain point (3) two are continuous at a certain point Their quotients are still functions that are continuous at this point, as long as the function value of the denominator at this point is not equal to 0.
Prove (3)
Let f(x), g(x) be continuous at the point x0,
then there is lim(x ->x0) f(x) = f(x0),lim(x->x0)=g(x0) <>0
Prove the continuity of trigonometric functions
Example y=sinx , y=cosx at (-∞, +∞ ) is continuous in
y=tanx, y=cotx is continuous in its domain.
Prove that tangent and cotangent function
y = tanx = sinx/cosx
y = cotx = cosx/sinx
Using the continuity of the continuous function quotient, it can be seen that tanx and cotx are in continuous within the domain of definition.
Exercises 1, 2, 3, 5, 6, 7
2 Continuity of inverse and composite functions
If function y = f(x) is continuous on the interval Ix, (monotonically increasing or monotonically decreasing), then its inverse x=φ(y) must exist
, and it is also monotonically increasing (decreasing) and continuous on the corresponding interval Iy = {y|y=f(x), x belongs to Ix}
For example: y=sinx is monotonically increasing and continuous on Ix [-π/2,π/2] , so its inverse function y=arcsinx is monotonically increasing on [-1,1] and is continuous.
y=cosx is monotonically decreasing on Ix[0,π], so its inverse function is monotonically decreasing in the interval [-1,+1] and continuous
y=tanx is monotonically decreasing in (-π/2,π/2) increases and is continuous, its inverse function is monotonically increasing in y=arctanx in the (-∞, +∞) interval and continuous
y=cotx is monotonically decreasing and continuous in (0, π), then its inverse function y=arccotx is in ( -∞, +∞) monotonically decreasing and continuous
(2) Let when x->x0, u=φ(x) limit exists, and lim(x->x0) φ(x) = a, and y = f (u) is continuous at the corresponding point u= a,
then when x->x0, the limit of the review function f[φ(x)] exists, and lim(x->x0)f[φ(x)] =
Proof of f(a) :
For example: Find lim(x->0) ln(1+x)/x