This is a good post. I remember being so confused in a real analysis class when my professor started talking about how important it is that we restrict our attention to continuous linear functions (what on Earth was a discontinuous linear function supposed to be?). This post explains what’s going on better than my professor or textbook did.
I agree with one of the other commenters that this part is not technically phrased accurately:
One way to think about continuity is that a function not having any “jumps” implies that it can’t have an “infinitely steep” slope
Because eg the derivative of f(x)=3√x at x=0 is +∞ despite the fact that it’s continuous there.
This is a good post. I remember being so confused in a real analysis class when my professor started talking about how important it is that we restrict our attention to continuous linear functions (what on Earth was a discontinuous linear function supposed to be?). This post explains what’s going on better than my professor or textbook did.
I agree with one of the other commenters that this part is not technically phrased accurately:
Because eg the derivative of f(x)=3√x at x=0 is +∞ despite the fact that it’s continuous there.