# Continuous Functions

Continuous functions are a fundamental concept in mathematics, playing a pivotal role in various fields such as calculus, analysis, and topology. A function is considered continuous if its graph has no abrupt jumps, breaks, or holes, exhibiting a smooth and uninterrupted behavior. Formally, a function f(x) is continuous at a point if, intuitively, small changes in the input correspond to small changes in the output without sudden fluctuations. Understanding the properties and behavior of continuous functions is crucial for solving problems across diverse mathematical disciplines, making them a cornerstone of mathematical theory and application.

Questions

- How do you show that the function #f(x)=1-sqrt(1-x^2)# is continuous on the interval [-1,1]?
- Is the following function continuous at #x=3# ?
- Is it correct to write #1^@=2pi/360^@#? Yes or no, and also why?
- 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?
- Is my answer correct?
- Show that any linear function #f(x) = ax + b# is a continuous function in any #x_o in ℝ# ?
- Find #lim_{x to oo} {pi/2 - arctan(x)}^(1/x)# ?
- Calculate f prime for the following continuous function?
- Hello all, Can please someone help me with this question? with plenty or sufficient details please.Any kind of help would be greatly appreciated. Thank you.
- For what value of the constants A and B is the function (ax+3) continuous for all X if x > 5?
- Lim (X^3+8/x+2). ? X=-2
- 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?
- Determine the value of the constant a if the function #f(x)# defined below is continuous at #x=2#. #f(x)={(ax^2+7x; x≤2),(3x^2+3a; x>2):}# ?
- Give an example which is continous everywhere but not differentiable at 3 points?
- Let f be a function defined by: ?
- If the function #f(x) = (1 - x)tan(pix/2)# is continuous at #x = 1#, then #f(1)# is ??
- F(x) = (x cosx + 3 tanx)/ (x² + sinx), For x≠0 f(x) = 4, for x=0 At x=0 is such that? (a) it is continous (b) it has irremovable discontinuity (c) it has removable discontinuity (d) lim f(x) = 3 x->0
- F(x)= {(ax-1)^3 if x>2 F(x)= {(ax-1)^3 if x=2 F(x)={(ax^2-1 if x<2 determine the real values of a ?
- If #f(x)={(ln3x", " 0<x<=3),(xln3", "3<x<=4) :}#, then #lim_(xto3)f(x)# is?
- Let f be given by the formula?