M171 Calculus I · Unit 2: Computing derivatives · on Exam 2
§1.7 Limits, continuity, and differentiability
Study notes for Limits, continuity, and differentiability in M171 Calculus I at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- $f$ is continuous at $a$ when three things hold: $f(a)$ is defined, $\lim_{x\to a}f(x)$ exists, and they are equal. Informally: no hole, no jump, no vertical asymptote at $a$.
- Discontinuities: removable (hole), jump (one-sided limits differ), infinite (vertical asymptote).
- $f$ is differentiable at $a$ when $f'(a)$ exists (the difference-quotient limit). Not differentiable at a corner, a cusp, a vertical tangent, or any discontinuity.
- Differentiable $\Rightarrow$ continuous. The converse is false: $|x|$ is continuous but not differentiable at $0$.
- To make a piecewise function continuous at the break, set the two one-sided limits equal and solve for the unknown constant.
Key formulas
Continuity at a
$$\lim_{x\to a} f(x) = f(a)$$
Implication
$$\text{differentiable at } a \;\Longrightarrow\; \text{continuous at } a$$
Worked example
Find $k$ so that $f(x)=\begin{cases} x^2 + k & x<2 \\ 3x - 1 & x\ge 2\end{cases}$ is continuous at $x=2$.
Show the solution
Left limit: $4 + k$. Right limit and value: $3(2)-1 = 5$. Continuity needs $4+k=5$, so $k=1$.
Common mistakes
- Claiming continuous implies differentiable.
- Checking only that the limit exists, without comparing it to $f(a)$.
Exam tipWhen asked "is f differentiable at a?", first check continuity; if it fails, you are done.
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