Continuous Functions

Key Questions

• How can I prove that a function is continuous?

  You can prove that a function `f(x)` is continuous at `a` by verifying that `lim_{x to a}f(x)=f(a)`.

  Wataru · 1 · Oct 10 2014
• Is the function `(x^2-6x+9)/(x-3)` continuous?

  No, it is not continuous at `x=3` since it is not defined there (zero denominator).

  Wataru · · Oct 10 2014
• What are continuous functions?

  Answer:

  We may also state two alternative definitions of continuous functions, using either the sequential criterion or else using topology and open sets.

  Explanation:

  Alternative definition number 1
  Let `f: X ->Y` be a function and let `(x_n)` be a sequence in X converging to an element x in X, ie `lim(x_n)=x in X`
  Then f is continuous at x iff and only if the sequence of function values converge to the image of x undr f, ie `iff lim (f(x_n))=f(x) in Y`

  Alternative definition number 2
  Let `f: X ->Y` be a function. Then f is continuous if the inverse image maps open subsets of Y into open subsets in X.
  ie, `AAA_(open)subeY =>f^(-1)(A)` is open in X

  Trevor Ryan. · 1 · Sep 17 2015

• What are continuous functions?
• What facts about continuous functions should be proved?
• How do you use continuity to evaluate the limit `sin(x+sinx)` as x approaches pi?
• How do you find values of x for which the function `g(x) = (sin(x^20+5) )^{1/3}` is continuous?
• How do you find values of x where the function `f(x)=sqrt(x^2 - 2x)` is continuous?
• How do you use continuity to evaluate the limit sin(x+sinx)?
• Given two graphs of piecewise functions f(x) and g(x), how do you know whether f[g(x)] and g[f(x)] are continuous at 0?
• How do you find the interval notation to prove `f(x)= x/(sqrt(1-x^2))` is continuous?
• How do you use continuity to evaluate the limit `(e^(x^2) - e^(-y^2)) / (x + y)` as `(xy)` approached `(1,1)`?
• How do you show that the function `f(x)=1-sqrt(1-x^2)` is continuous on the interval [-1,1]?
• Is it possible for a function to be continuous at all points in its domain and also have a one-sided limit equal to +infinite at some point?
• At what point is `f(x) = x - [x]` discontinuous?
• How do you show the function is continuous at the given number a `f(x)=(x+2x^3)^4`, a=-1?
• Is a function differentiable at all points that it is continuous?
• How do you use the definition of continuity and the properties of limits to show that the function `f(x) = 5x^4 - 9x^3 + x - 7` is continuous at a given number a=7?
• How do you use the definition of continuity and the properties of limits to show the function is continuous `F(x)= x+sqrt(x-1)` on the interval [1, inf)?
• How do you use the definition of continuity and the properties of limits to show the function is continuous `F(x)= (x^2-8)^8` on the interval (-inf, inf)?
• How do you use the definition of continuity and the properties of limits to show that the function `f(x)=x^2 + sqrt(7-x)` is continuous at a=4?
• How do you use the epsilon-delta definition of continuity to prove `f(x) = x^2` is continuous?
• How do you use the definition of continuity and the properties of limits to show that the function `h(t)=(2t-3t^2)/(1+t^3)` is continuous at the given number a=1?
• How do you use the (analysis) definition of continuity to prove the following function `f(x) = 3x +5` is continuous for all x in R?
• By the definition of continuity, how do you show that `xsin(1/x)` is continuous at x=0?
• How do you use the definition of continuity to determine if `g(x) = x^3 / x` is continuous at x=0?
• Question #a72ca
• Is it correct to write `1^@=2pi/360^@`? Yes or no, and also why?
• Question #68e60
• Question #9387b
• Is the graph of `f (x) = (X^2 + x)/x` continuous on the interval [-4, 4]?
• For what value of the constants A and B is the function (ax+3) continuous for all X if x > 5?
• For what value of the constants A and B is the function f(x)=8 continuous for all X if x = 5?
• For what value of the constants A and B is the function `f(x)=x^2+bx+1` continuous for all X if x < 5?
• If g(x)=x if x<0, x^2 if `0<= x <=1`, x^3 if x>1, how do you show that g is continuous on (all real #'s)?
• What are the three conditions necessary for the function f(x) to be continuous at the point x = c?
• What are the the three things a continuous function can't have?
• How do you find the value of k where the 2 function continuous `g(x)= |x^3|` for `-oo<x<k` and `x^2` for `k<x<oo`?
• How do you find the values of 'a' and 'b' that make f(x) continuous everywhere given `(x^2 - 4)/(x - 2)` if x < 2, `ax^2 - bx + 3` if 2 < x < 3 and `2x - a + b` if `x>=3`?
• For what value(s) of k is the function f(x) continuous at x = -3 given `f(x) = -6x - 12` when x < -3, `f(x) = k^2 - 5k` when x = -3 and f(x) = 6 when x > -3?
• If f(x) is continuous and 0 <= f(x) <= 1 for all x on the interval [0,1], then is it true that for some number x, f(x) = x?
• Why the function is discontinuous at a=1 given `f(x)= 1-x^2` if x < 1 and `f(x)= 1/x` if `x >= 1`?
• How do you find the intervals on which the function is continuous given `y = (2)/((x + 4)^2) + 8`?
• How do you find the intervals on which the function is continuous given `y = sqrt(5x + 9)`?
• How do you find the intervals on which the function is continuous given `y = ln(3x-1)`?
• Is the function f(x) = secx continuous on the interval [-pi/2, pi/2]?
• If `f(x)=(x^2+6x+8)/(x+2)` if x<-2 and `kx^2`if `x>= -2`, what value of k would make the function continuous at x=-2?
• How do we find the values of k and m that makes function continue anywhere piecewise function of `(x^2) + 5` when x > 2, `m(x+3) + k` when `-1 < x <=2` and `2(x^3) + x + 7` when `x <=-1`?
• How do you find the values of c and d that make the following function continuous for all x given `f(x) = 9x` if x<1, `f(x) = cx^2+d` if `1<=x<2` and `f(x) = 3x` if `x>=2`?
• How do you find the constant a so that the function is continuous on the entire real line given `f(x)=4,`x <= -1`, ax +b, -1< x <1 and 6,`x>=1#?
• How many continuous functions f are there which satisfy the equation `(f(x))^2 = x^2` for all x?
• How do you explain why `f(x) = x^2 +1` is continuous at x = 2?
• How do you explain why `f(x) = x^2` if `x>=1` and 2x if x<1 is continuous at x = 1?
• How do you prove that the function: `T(x) = 1 / (abs(x-2)-x^2)` is continuous between [1.5,8]?
• Consider the function defined by f(x)= sinx/x if `-1 <=x < 0`, ax+b if `0 < =x <= 1` and `(1-cosx)/x` if x >1, how do you determine a and b such that f(x) is continuous on [-1,2]?
• Is the function `2x-x^2` continuous if 0 ≤ x ≤ 2?
• Is the function `2-x` continuous if 2 < x ≤ 3?
• Is the function `x-4` continuous if 3 < x < 4?
• How do you use the definition of continuity to determine weather f is continuous at `f(x)= x-4` if `x<=0` and `x^2+x-4` if x>0?
• How do you use the definition of continuity to determine weather f is continuous at `2-x` if x<1, 1 if x=1 and `x^2` if x>1?
• Judge the following is true or false If f is continuous on ( 0,1 ) then there is a c in (0,1) such that f(c) is a maximum value of f on (0,1)?
• How do I find all intersection points of `f(x)=ax^4` and `g(x)=bx^2` where a,b > 0?
• How do you show that the function `h(x)=xe^sinx` is continuous on its domain and what is the domain?
• How do you show that the function `g(x)=sqrt(x^2-9)/(x^2-2)` is continuous on its domain and what is the domain?
• What is the continuity of the composite function `f(g(x))` given `f(x)=x^2` and `g(x)=x-1`?
• What is the continuity of the composite function `f(g(x))` given `f(x)=1/sqrtx` and `g(x)=x-1`?
• What is the continuity of the composite function `f(g(x))` given `f(x)=1/(x-6)` and `g(x)=x^2+5`?
• What is the continuity of the composite function `f(g(x))` given `f(x)=sinx` and `g(x)=x^2`?
• Question #bc73c
• Question #29a12
• Over what domain is `f(x)=sin(x)` continuous ?
• Question #b1bc4
• Question #04e10
• Question #4cbca
• How do you prove that `sqrtx` is continuous?
• The function `F(x)= x^( 2/3)` is continuous everywhere. How do you find the maximum and minimum values on [-1,2]?
• Question #36d91
• Question #f7666
• How do you find the factorial of negative numbers?
• Question #11fb2
• Question #a6851
• `lim_(x->0)(tan x - x)/x^3 =` ?
• Is the following function continuous at `x=3` ?
• Suppose `f(x)` is a real-valued function, defined and continuous on `[0, 2]` with `f(0) = -1`, `f(1) = 1` and `f(2) = -1`. Which of the following statements are true?
• Question #b2764
• Given `f(x) = sint/t`. How do you show that `f(t)` has a maximum value as `t->0`?
• Find `a` and `b` so that `f(x) = { (x^2, xle1),(ax+b,x gt 1) :}` is continuous?
• How to solve this? Determine the continuity and derivability domain for `f(x)` `f:[0,oo)->RR,f(x)=|x-1|sqrtx`
• Determine all real possible s, so that `W={f|int_-1^1f(x)dx=s}` is subspace in real function that continue in `[-1,1]` ?
• Give an example which is continous everywhere but not differentiable at 3 points?
• How to prove that `f(x)=|x|` is continue at `0` ?
• Question #34fda
• Find the values of m and b that make f continuous everywhere: m = ? b = ?
• Question #59ddc
• Which values of x this function is continuous? (x non-negative real number and n->infinity)
• Show that f(x)=2a+3b is continous, where a and b are constants?