Lesson 3 of 13 · 7 min
Algebra of continuous functions
NCERT §5.2.1
Riya's daily cost is the petrol cost plus the parking fee. If each part changes smoothly over some stretch of time, must their sum change smoothly too?
The lesson in notes
In short
If f and g are both continuous at c, then f + g, f − g and f·g are continuous at c, and f/g is too provided g(c) ≠ 0. The proofs follow from the matching rules for limits.
Special cases: λf is continuous for any constant λ (so −f is too), and 1/g is continuous wherever g ≠ 0.
Every rational function p(x)/q(x) is continuous at every point where q(x) ≠ 0, since polynomials are continuous.
sin x and cos x are continuous everywhere, so tan x = sin x/cos x is continuous at every point of its domain, that is wherever cos x ≠ 0.
Composites: if g is continuous at c and f is continuous at g(c), then f∘g is continuous at c. So sin(x²) is continuous, being sine applied to the continuous x².
Worked example: |1 − x + |x|| is continuous for every real x, as the modulus of the sum of a polynomial and |x|.
The rules only give sufficient conditions. A sum of two discontinuous functions can still be continuous.