# Definition of Continuity at a Point

The concept of continuity at a point is fundamental in the study of calculus and real analysis. It refers to the smoothness or uninterrupted nature of a function at a specific point in its domain. In mathematical terms, a function is said to be continuous at a point if the limit of the function exists at that point and is equal to the value of the function at that point. Understanding continuity at a point is crucial for analyzing the behavior of functions and solving various mathematical problems.

Questions

- How do you prove that the function #f(x)=(x-5)/ |x-5|# is continuous everywhere but x=5?
- How do you prove by definition that the function #f(x)= x^2 sin (1/x)# is continuous at x=0?
- How do you find the points of continuity and the points of discontinuity for a function?
- How do you prove that the function #f(x) = [x^2 + x] / [x]# is not continuous at a =0?
- How do you prove that the function #1/x# is continuous at x=1?
- What is the definition of continuity at a point?
- How do you prove that the function #f(x) = x^2 -3x +5# is continuous at a =2?
- How do you prove that #g(x) = 1/2x# is continuous at #x=1/4#?
- For a continuous function (let's say f(x)) at a point x=c, is f(c) the limit of the function as x tends to c? Please explain.
- How do you prove that the function #f(x)=(x^2+7x+10)/(x+5) # is continuous everywhere but x=-5?
- For what intervals is #f(x) = tan((pix)/4)# continuous?
- If #f(x)= (x^2-9)/(x+3)# is continuous at #x= -3#, then what is #f(-3)#?
- How do you use continuity to evaluate the limit #arctan(x^2-4)/(3x^2-6x)#?
- How do you find where #f(x)=1/ln(x)# is continuous and differentiable on which interval(s)?
- How do you prove that the function #g(x) = x^3 / x# is continuous everywhere but x=0?
- How do you find the points of continuity of a function?
- What does continuity mean?
- Let f be a function so that (below). Which must be true? I. f is continuous at x=2 II. f is differentiable at x=2 III. The derivative of f is continuous at x=2 (A) I (B) II (C) I & II (D) I & III (E) II & III
- What are the three conditions for continuity at a point?
- What does continuous at a point mean?