MathSquareRoo said:Homework Statement
Prove that if f is uniformly continuous on [a,b] and on [a,c] implies that f is uniformly continuous on [a,c].
Homework Equations
The Attempt at a Solution
This is my rough idea for a proof, can someone help be say this more formally? Is my thinking even correct?
Let epsilon > 0.
Then there is some delta_1 for [a,b] and some delta_2 for [b,c].
Then the minimum of delta_1 and delta_2 is the delta we want for [a,c].
MathSquareRoo said:Thanks. What changes should I make to make it more formal?
Continuity in real analysis refers to a mathematical property of a function where there are no abrupt changes or discontinuities in its graph. This means that the function is smooth and can be drawn without lifting the pen from the paper.
While both continuity and differentiability are properties of functions, they are distinct concepts. Continuity focuses on the behavior of a function at a point, while differentiability looks at how the function changes between points. A function can be continuous but not differentiable, but it cannot be differentiable without being continuous.
Continuity is crucial in real analysis as it allows us to study the behavior of functions in a smooth and predictable manner. It enables us to analyze functions and their derivatives, which are essential in many mathematical applications, such as optimization, physics, and engineering.
A function is continuous at a point if the limit of the function at that point exists and is equal to the value of the function at that point. To determine if a function is continuous on an interval, we need to check the continuity at each point in the interval.
Yes, a function can be discontinuous at a single point. This occurs when the limit of the function at that point does not exist or is not equal to the value of the function at that point. This is known as a point discontinuity or a removable discontinuity.