Is derivative continuous?

Is a Derivative Always Continuous? Let’s Get Real About Calculus’s Quirks

So, you’re diving into calculus, huh? You’re probably wondering about derivatives – those magical tools that tell us how a function is changing. A question that often pops up is this: if a function has a derivative, does that mean the derivative itself is always smooth and continuous? The short answer? Nope. And that’s where things get interesting, revealing some of the beautiful weirdness hidden within calculus.

Let’s rewind a bit and talk about differentiability and continuity. Think of it this way: for a function to have a derivative at a specific point, it needs to be, well, nice at that point. Mathematically, we’re talking about the limit of that difference quotient thing existing. But in plain English, it means you can draw a tangent line.

Now, here’s a key concept: If a function is differentiable, it has to be continuous. Imagine trying to draw a tangent line on a graph that suddenly jumps or breaks – impossible, right? That’s why differentiability implies continuity.

But here’s the kicker: just because a function is continuous doesn’t automatically mean it’s differentiable. Remember the absolute value function, f(x) = |x|? It’s continuous at x = 0, but try drawing a smooth tangent line at that sharp corner. Can’t be done!

Okay, so continuous functions aren’t always differentiable. But what about a function that is differentiable? Can its derivative still be a bit of a rebel and be discontinuous? Believe it or not, yes! These functions are a bit rarer, not something you usually run into in your first calculus class, but they’re out there.

The classic example? Buckle up, it’s a bit of a mouthful:

f(x) = x²sin(1/x) if x ≠ 0


f(x) = 0 if x = 0

This function is differentiable everywhere, even at x = 0. Trust me on this (or work it out yourself – it’s a good exercise!). But when you find its derivative, things get a little crazy:

f'(x) = 2xsin(1/x) – cos(1/x) if x ≠ 0


f'(x) = 0 if x = 0

The derivative exists for all x, but it’s not continuous at x = 0. That pesky cos(1/x) term goes wild as x gets closer and closer to zero, oscillating like crazy. It’s like a toddler who’s had too much sugar, preventing the derivative from settling down and having a limit at that point. This creates what’s called an essential discontinuity. Fancy, right?

So, what’s going on here? The x² term is like a gentle hand, squeezing those wild oscillations of sin(1/x) just enough to force the function to be differentiable at zero. But the derivative still has that cos(1/x) term, which is just too jittery to let the derivative be continuous.

Why should you care? Well, it shows that differentiability is a stronger condition than the continuity of the derivative. We even have a name for functions with continuous derivatives: continuously differentiable. It’s a distinction that matters when you get into more advanced math.

These functions might seem like weird edge cases, but they pop up in some unexpected places in math. Volterra’s function, for instance, has a derivative that’s discontinuous on a set of positive measure. (Don’t worry too much about what that means right now!)

The takeaway? Just because a function is differentiable doesn’t mean its derivative is automatically well-behaved. The example of x²sin(1/x) is a reminder that calculus can be full of surprises, and that’s part of what makes it so fascinating. Keep exploring, and you’ll uncover even more cool quirks along the way!