acacia89 said:Suppose a function f : R → R satisfy f(x + y) = f(x) + f(y) and f is continuous
at x = 0: Show that f is continuous at every point in R.
(Hint: Using the fact that
lim f(x) = l implies
x→x0
limf(x0+h)= l
h→0 )
In mathematics, continuity is a property of a function where small changes in the input result in small changes in the output. In other words, a function is continuous if there are no abrupt jumps or breaks in its graph.
To prove that a function f is continuous at a point a, we need to show that the limit of f(x) as x approaches a is equal to f(a). This can be done by evaluating both the left and right limits of f(x) at a and showing that they are equal.
Proving that a function is continuous at every point in R (the set of real numbers) is important because it guarantees that the function is well-behaved and predictable. It allows us to make accurate predictions and perform operations, such as integration and differentiation, on the function.
Yes, a function can be continuous at some points and not others. For example, a function may be continuous on a closed interval [a, b] but not at the endpoints a and b. This is known as a piecewise continuous function.
There are three types of continuity: pointwise continuity, uniform continuity, and differentiability. Pointwise continuity means a function is continuous at every point in its domain. Uniform continuity means a function is continuous across its entire domain, with the size of the input not affecting the size of the output. Differentiability means a function can be differentiated at every point in its domain.