# What's an example of a function where both Left & Right hand limits are finite, but not equal?

Can someone help me find a function where both the LHL & RHL are finite numbers (such as 1, 2, 3....etc), but not equal to each other?

For example, consider f(x) = |x| - x. When x -> 0, we know that LHL = 0 and RHL is also equal to 0.

Is there a function where these are unequal?

EDIT: Note when I say function, I do not mean just graph. I mean it formulated as f(x) = something

• Where are you stuck? Could you draw an example? What would it look like? Jul 8, 2023 at 4:37
• As a hint, a formula like $f(x) = |x| - x$ is not necessary to define a function. The previous comment suggesting that you draw an example is an even better hint. Jul 8, 2023 at 4:42
• It sounds like you're confusing "function" with "formula". Not all functions can be expressed symbolically. But the floor function $f(x)=\lfloor x \rfloor$ is an example.
– Karl
Jul 8, 2023 at 4:51
• You're going to struggle a bit if you're trying to build a function from components that are continuous through addition, multiplication or composition because those typically preserve continuity. You're going to have to use a function that's already defined as discontinuous - like the suggested floor function, or a Heaviside step function or signum function. Jul 8, 2023 at 4:58
• I figured eventually we needed to get to a formal definition, not just a picture. Have your students seen piecewise functions before? What about the greatest integer function? Jul 8, 2023 at 5:00

So for a function to have both sided limits defined, but them not being equal to each other will result in a discontinuity at that point.

The floor function or the greatest integer (less-than or equal to) function are good examples of such a function.

Picture Source: https://mathworld.wolfram.com/FloorFunction.html

Consider the above function f(x) = ⌊𝑥⌋

When x approaches 0 from the left:

$$\lim_{x \to -0} f(x)$$ = -1

But when x approaches 0 from the right:

$$\lim_{x \to +0} f(x)$$ = 0

In this case, both LHL and RHL are finite numbers , but $$LHL \ne RHL$$

• It's been several decades, but this may very well be the first function I saw in school that had distinct RH and LH limits. Jul 8, 2023 at 5:18