Show that f(x)={ x^2+2, x ≤ 1 x+2, x>1 . is continuous but not differentiable at x=1

Show that f(x)={ x^2+2, x ≤ 1 x+2, x>1 . is...

Key Concepts

Continuity

Continuity describes a function's property where there are no 'jumps' or 'holes' at a particular point. More formally, a function is continuous at a point if the limit as the variable approaches that point exists and equals the function’s value at that point. In problems like these, verifying continuity at the boundary involves showing that the left-hand and right-hand limits agree with each other and with the function’s assigned value.

Piecewise Functions

Piecewise functions are defined by different expressions in different parts of their domain. When dealing with such functions, special attention must be paid at the points where the defining expressions change. Analyzing these points often requires checking both continuity and differentiability by separately examining the behavior of the function on either side of the transition point.

Limits

Limits are fundamental to both continuity and differentiability. They mathematically capture the behavior of a function as the input approaches a specific point. In evaluating the continuity at the transition between pieces of a piecewise-defined function, one computes the limits from the left and the right, ensuring they coincide with the function's value at that point.

Differentiability

Differentiability at a point requires that the function has a well-defined tangent line or derivative there. For a function to be differentiable, not only must it be continuous, but its rate of change must also approach the same value from both sides. In cases where a function is defined piecewise, ensuring differentiability at the boundary involves verifying that the left-hand and right-hand derivatives are equal.

One-sided Derivatives

One-sided derivatives refer to the derivatives computed from the left and the right of a specific point. In the context of piecewise-defined functions, especially at the boundary points, the existence of both the left-hand and right-hand derivatives and their equality is essential for overall differentiability at that point. A mismatch between these derivatives results in the function not being differentiable, even if it remains continuous.

Transcript

00:01 In this problem, we're given f of x equals x squared plus 2 or x plus 2.

00:11 So the first case is x less than equal to 1.

00:14 The second case is x greater than 1.

00:18 So what we want to do is show f is continuous but not differentiable.

00:32 Forgive my rating.

00:33 Okay.

00:33 So at x equals 1, at x equals 1.

00:38 So how do we show continuity? so we want to show that limit as x approaches a of f of x equals f of a, or a equals 1.

00:53 So the limit as x approaches 1 of f of x is going to, so let's look at the left -hand side.

01:03 So from the left -hand side, that's going to equal the limit as x approaches x.

01:11 Approaches 1 of x squared plus 2 so that's gonna this is a polynomial we can evaluate it directly so we get 1 plus 2 is 3 and the limit as x approaches 1 from the right -hand side of f of x is going to be x plus 2 so that equals 1 plus 2 is 3 these agree therefore f is continuous at a equals 1 so next we will want to look at the drift.

01:54 So what's the definition of the derivative? so the limit as x approaches h of f of x plus h minus f of x over h.

02:05 So it's going to equal.

02:09 Okay, so let's look at the two sides.

02:11 So actually this should be x as h approaches zero.

02:15 So let's look at it from the left -hand side and the right -hand side.

02:18 Left -hand side of zero and right -hand side.

02:22 So on left -hand side of zero, we are going to be looking at the limit as x plus h squared plus 2 minus x squared equal to.

02:45 So it's going to be, oh, we can just use a.

02:47 That makes more sense.

02:49 Let's look at a instead of x.

02:54 So a was equal to one.

02:57 So we'll look at the left hand side.