Calculus Continuity

We’ve already internalised the idea of Limits. They’re pretty obvious; “What does the value of a function equal when the input of that function gets super close to another arbitrary value a.”

Continuity is another quite simple idea. Basically, that a line (or output of a function), along a given range, is unbroken.

Continuity

There’s 3 requirements for a function $f(x)$ (within a range) to be continuous.

• $f(a)$ exists
• $li{m}_{x\to a}(f(x))$ exists
• $li{m}_{x\to a}(f(x))=f(a)$

Essentially, a function has an output $L$ at a given point, and when you approach it from either side, it looks like the limit is that same $L$.

Important!

Differentiability implies continuity